On the Tate conjecture for divisors on varieties with ℎ2,0 = 1 in positive characteristics

Paul Hamacher; Ziquan Yang; Xiaolei Zhao

doi:10.1515/crelle-2025-0042

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On the Tate conjecture for divisors on varieties with ℎ^2,0 = 1 in positive characteristics

Paul Hamacher , Ziquan Yang and Xiaolei Zhao

Published/Copyright: July 10, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 827

Abstract

We prove that the Tate conjecture for divisors is “generically true” for mod ⁡ p reductions of complex projective varieties with h 2 , 0 = 1 , under a mild assumption on moduli. By refining this general result, we establish a new case of the BSD conjecture over global function fields, and the Tate conjecture for a class of general type surfaces of geometric genus 1.

1 Introduction

In this paper, we prove a general theorem stating that the Tate conjecture for divisors is “generically true” for fibers in an arithmetic family of varieties with h 2 , 0 = 1 , under some mild assumptions, and then we provide a method to refine this theorem in concrete situations. To motivate the discussion, we begin with an example of particular interest.

For an elliptic curve ℰ over a global field, (the basic version of) the BSD conjecture predicts that the analytic rank is equal to the Mordell–Weil rank. When the analytic rank of is at most 1, the conjecture is known in many cases over number fields (e.g., [37]) and in full generality over function fields [60, 53]. When the analytic rank is greater than 1, much less is known, even over function fields. However, a special feature of function fields is that the BSD conjecture is equivalent to the Tate conjecture for the corresponding elliptic surfaces [31, 44], so one can alternatively attack through the Tate conjecture. As a refinement of our general theorem, we provide a new class of examples over function fields.

Theorem A

Let 𝐶 be a smooth projective curve over a finite field 𝑘 of characteristic p ≥ 5 , let ℰ be an elliptic curve over the function field k ⁢ ( C ) , and let π : X → C be the corresponding minimal elliptic surface. If g ⁢ ( C ) = h ⁢ ( E ) = 1 , and all geometric fibers of 𝜋 are irreducible, then the BSD conjecture holds for ℰ, i.e., rank ⁡ E ⁢ ( k ⁢ ( C ) ) = ord s = 1 ⁢ L ⁢ ( E , s ) .

Here g ⁢ ( C ) denotes the genus of 𝐶, and h ⁢ ( E ) denotes the height of ℰ (i.e., the degree of the fundamental line bundle L : = ( R 1 π ∗ O X ) ∨ ). The above theorem is analogous to the classical result of Artin and Swinnerton-Dyer [4] for elliptic K3 surfaces, which correspond to the case when g ⁢ ( C ) = 0 and h ⁢ ( E ) = 2 . We remark that our elliptic surfaces have Kodaira dimension 1, and the highest analytic rank achieved by the corresponding elliptic curves is 10 (see Proposition 7.4.6). The condition that all geometric fibers of 𝜋 are irreducible is saying that the Weierstrass normal form of 𝑋 is smooth. In the sequel of this paper by the second author and Guo [24], this condition is dropped in many cases, and completely removed when p ≥ 11 .

Now we introduce our general theorem. We shall consider arithmetic families of the following type. Suppose that 𝐾 is a field of characteristic 0 and 𝑆 is a smooth connected 𝐾-variety. We say that a smooth projective morphism f : X → S is a ♡ -family if, for every geometric point s → S , the fiber X s is connected, dim H 0 ⁢ ( Ω X s 2 ) = 1 (i.e., h 2 , 0 ⁢ ( X s ) = 1 ), and the Kodaira–Spencer map

∇ : T S / K → Hom ¯ ⁢ ( R 1 ⁢ f ∗ ⁢ Ω X / S 1 , R 2 ⁢ f ∗ ⁢ O X )

is nontrivial. Our general theorem states the following.

Theorem B

Let 𝖬 be a connected and separated scheme over Spec ⁢ ( Z ) which is smooth and of finite type and let f : X → M be a smooth projective morphism with geometrically connected fibers. If the restriction of 𝒳 to M Q is a ♡ -family, then for p ≫ 0 , every fiber of 𝒳 over a point s ∈ M F p satisfies the Tate conjecture for divisors, i.e., for every prime ℓ ≠ p , the Chern class map

c 1 : Pic ⁡ ( X s ) ⊗ Q ℓ → H ét 2 ⁢ ( X s ̄ , Q ℓ ⁢ ( 1 ) ) Gal ⁢ ( s ̄ / s )

is surjective, where s ̄ is a geometric point over 𝑠.^[1]

Clearly, it is both a positive characteristic analogue of Moonen’s main result in [49], and a generalization of the Tate conjecture for K3 surfaces, for which people made great progress in the past decade (e.g., [40, 42, 9]). The base scheme 𝖬 above should be thought of as the moduli of varieties of a certain type. Note that we do not require in the theorem above that 𝑠 is a closed point – it is allowed to have positive transcendence degree over F p . Also, although the above theorem is non-effective in 𝑝, for concretely given families, we often can make it effective, as in the case of Theorem A. To illustrate this, we also analyze a class of surfaces of general type.

Theorem C

Assume that 𝑘 is a field finitely generated over F p for p ≥ 5 . Let 𝑋 be a minimal smooth projective geometrically connected surface over 𝑘. Let K X denote its canonical divisor, p g the geometric genus h 0 ⁢ ( K X ) , and 𝑞 the irregularity h 1 ⁢ ( O X ) . If p g = K X 2 = 1 , q = 0 , and K X is ample, then 𝑋 satisfies the Tate conjecture.

We remark that the condition “all geometric fibers of 𝜋 are irreducible” in Theorem A is an analogue of the condition “ K X is ample” above. Over ℂ, surfaces with the above invariants were classified by Todorov and Catanese [59, 7]. They are simply connected, have a coarse moduli space of dimension 18, and were among the first examples of p g = 1 surfaces for which both the local and the global Torelli theorems fail [8].

Sketch of proofs

We first explain how to prove Theorem B. We build on the overall strategy of Madapusi-Pera [40], which has two main steps. The first is to construct an integral period morphism ρ : M → S , where 𝒮 is the canonical integral model of a Shimura variety Sh ⁢ ( G ) defined by a suitable special orthogonal group 𝐺.^[2] Up to passing to its spinor cover, 𝒮 is equipped with a family of abelian schemes 𝒜. The second is to construct, for each geometric point s → M and t = ρ ⁢ ( s ) , a morphism θ : L ⁢ ( A t ) → NS ⁢ ( X s ) , where L ⁢ ( A t ) is a distinguished subspace of End ⁡ ( A t ) . Then the Tate conjecture for NS ⁢ ( X s ) follows from a variant of Tate’s theorem for L ⁢ ( A t ) . As a crucial step, Madapusi-Pera proved that 𝜌 is étale. This boils down to the geometric fact that a K3 surface 𝑋 has unobstructed deformation and H 1 ⁢ ( Ω X ) ≅ H 1 ⁢ ( T X ) . Unfortunately, this is rarely true when 𝑋 is not a close relative of a hyper-Kähler variety.

The main contribution of our paper is a method to remove the dependence of the above strategy on any good local property of 𝜌 (not even flatness). Indeed, the condition on the Kodaira–Spencer map contained in the definition of a ♡ -family just amounts to asking

dim im ⁡ ( ρ C ) > 0 ,

so even the situation dim M < dim S is allowed in Theorem B. Below, we explain in more detail the difficulties in extending the two steps above and how to overcome them.

1.1

It is not hard to construct a period morphism ρ C : M C → Sh ⁢ ( G ) C over ℂ, as the target is a moduli of variations of Hodge structures with some additional data. As in [40], the idea to construct 𝜌 is to descend ρ C to a morphism ρ Q over ℚ, and then appeal to the extension property of 𝒮. If, for every s ∈ M ⁢ ( C ) , the motive h 2 ⁢ ( X s ) defined by H 2 ⁢ ( X s , Q ) is an abelian motive (e.g., in the K3 case), then one shows that the action of ρ C on ℂ-points is Aut ⁢ ( C ) -equivariant, so that ρ C descends to ℚ just as in [40].

To treat the general case, we absorb inputs from Moonen’s work [49]. Let b ∈ M ⁢ ( C ) be a point lying over the generic point of 𝖬, and let T ⁢ ( X b ) be the orthogonal complement of ( 0 , 0 ) -classes in H 2 ⁢ ( X b , Q ⁢ ( 1 ) ) . Let 𝐸 be the endomorphism algebra of the Hodge structure T ⁢ ( X b ) , which is known to be either a totally real or a CM field. In the latter case, one can still show that h 2 ⁢ ( X s ) is abelian for each 𝑠 (see Theorem 2.2.7), so that the argument of [40] still applies. In the former case, we need to consider an auxiliary Shimura subvariety Sh ⁢ ( G ) C ⊆ Sh ⁢ ( G ) C , defined by the Weil restriction 𝒢 of a special orthogonal group over 𝐸.

Up to replacing 𝖬 by a connected étale cover and 𝑏 by a lift, the restriction of ρ C to M ∘ factors through a morphism ϱ C : M ∘ → Sh ⁢ ( G ) C , where M ∘ is the connected component of M C containing 𝑏. We will show that the field of definition 𝐹 of M ∘ always contains 𝐸. Interestingly, to descend ρ C to ℚ, it suffices to descend ϱ C to 𝐹. The trick is to consider the left adjoint to the base change functor from ℚ-schemes to 𝐹-schemes. To show that ϱ C descends to 𝐹, consider the submotive t ⁢ ( X b ) of h 2 ⁢ ( X b ) defined by T ⁢ ( X b ) . When dim E T ⁢ ( X b ) is odd, although we do not know that t ⁢ ( X b ) (or equivalently h 2 ⁢ ( X b ) ) is abelian, Moonen’s work [49] tells us that (a slight variant of) “the E / Q -norm” Nm E / Q ⁢ ( t ⁢ ( X b ) ) is abelian. This allows us to show that ϱ C descends to 𝐹 because considering Nm E / Q ⁢ ( t ⁢ ( X b ) ) amounts to considering a different faithful representation of 𝒢. As we document in a separate reference file [63], under some hypotheses, the canonical models of Shimura varieties of abelian type over their reflex fields have a moduli interpretation attached to any faithful representation. When dim E T ⁢ ( X b ) is even, some adaptation is needed, which we omit here.

1.2

Next, we explain how to construct θ : L ⁢ ( A t ) → NS ⁢ ( X s ) , where the main novelty of our method lies. As in [40], the key to construct 𝜃 is to find, for each ζ ∈ L ⁢ ( A t ) , a characteristic 0 point t ̃ on 𝒮 lifting 𝑡, such that (i) 𝜁 deforms to A t ̃ and (ii) t ̃ comes from a lifting s ̃ of 𝑠 via 𝜌. The existence of t ̃ which satisfies (i) was already shown in [41]. When 𝜌 is étale (or at least smooth), (ii) is then automatically satisfied. This is where [40] crucially relies on the étaleness of 𝜌.

The challenge to generalize this, especially when dim M < dim S , is that there is no general way to characterize locally the image of 𝜌 in 𝒮, so for any given t ̃ , one cannot decide directly whether it satisfies (ii) or not. Indeed, this is essentially a local Schottky problem, which is famously hard. To overcome this, we revisit Deligne’s insight for [14, Theorem 1.6] but replace the local crystalline analysis by a global and topological argument.

Pretend for a moment that M C and M k are both connected, where 𝑘 is the (separably closed) field defining 𝑠, and set p = char ⁡ k . Let η ̄ and η ̄ p be geometric generic points over M C and M k respectively. Pick a prime ℓ ≠ p and restrict f : X → M to Z ( p ) . Suppose for the sake of contradiction that, for some 𝜁, there is no lifting t ̃ which satisfies (i) and (ii). Then using that the deformation of 𝜁 is controlled by a single equation, we can show that 𝜁 deforms along the formal completion of M k at 𝑠, and hence gives rise to an element of L ⁢ ( A ρ ⁢ ( η ̄ p ) ) . This further induces an element in ( R 2 ⁢ f ∗ ⁢ Q ℓ ) η ̄ p which is stabilized by an open subgroup of π 1 ét ⁢ ( M k , η ̄ p ) . On the other hand, we show using the theorem of the fixed part that all elements of ( R 2 ⁢ f ∗ ⁢ Q ℓ ) η ̄ stabilized by an open subgroup of π 1 ⁢ ( M C , η ̄ ) come from L ⁢ ( A η ̄ ) ⊗ Q ℓ . Therefore, to derive a contradiction, it suffices to show that M k does not have more “ π 1 -invariants” than M C .

Using Hironaka’s resolution of singularities and a spreading out argument, we can find an open subscheme 𝑈 of Spec ⁡ ( Z ) such that M U admits a compactification whose boundary is a relative normal crossing divisor. Then we apply Grothendieck’s specialization theorems for tame fundamental group and Abhyankar’s lemma to show that M k indeed cannot have more “ π 1 -invariants” than M C , when ( p ) ∈ U .^[3] This proves Theorem B.

1.3

To prove Theorems A and C, we need to avoid the spreading out argument above. Although compactifying moduli spaces is in general a hard geometric problem, one can find for 𝖬 in question a partial compactification, that is, a morphism M → B and a smooth proper 𝒫 over 𝐵 such that 𝖬 is an open subscheme of 𝒫. Then we can find lots of smooth proper curves in 𝒫. If the boundary D = P − M is generically reduced modulo 𝑝, then we can find a curve on M k which deforms to characteristic 0 such that, by looking at the curve, we can already prove that M k does not have more “ π 1 -invariants” than M C . Of course, such curve needs to be chosen wisely, and we do this by repeatedly applying the Baire category theorem. We will also need 𝜌 to satisfy a stronger condition, namely dim im ⁡ ( ρ C ) > dim B C , as opposed to just dim im ⁡ ( ρ C ) > 0 . This is known to hold for the surfaces in question.

The boundary 𝔇 is essentially a discriminant scheme, i.e., 𝒳 extends to a family over 𝒫, and 𝔇 is precisely the locus where the extension fails to be smooth. In general, it is possible for a discriminant scheme over ℤ to be generically non-reduced modulo a certain prime (cf. [55, Theorem 4.2]), so the task is to determine an effective range of 𝑝 for which this does not happen. Drawing ideas from enumerative geometry, we show that this happens only when a general fiber over D F ̄ p is “more singular” than that over D C . To exclude this possibility when p ≥ 5 for the surfaces in Theorem C, it suffices to adapt Katz’ results on Lefschetz pencils. Doing this for Theorem A is much more involved. In particular, we need to develop some nonlinear Bertini theorems (see Section 7.2) tailored to handle Weierstrass equations, the key input being Kodaira’s classification for the singular fibers in an elliptic fibration.

Remark 1.4

Recently, Fu and Moonen proved the Tate conjecture for Gushel–Mukai varieties in characteristic p ≥ 5 . The middle cohomology of these varieties behaves like that of a K3 surface up to a Tate twist. An earlier version of the paper also discussed these varieties. For our method, these varieties can be treated in a similar way to the surfaces in Theorem C. However, as Fu–Moonen [21] gave much more thorough treatment of these varieties and proved that the relevant integral period morphism 𝜌 is indeed smooth, we removed the section from the current version. In general, it is hard to determine whether 𝜌 is smooth integrally even when ρ C is smooth, as one can tell from [21], but when this can be achieved, 𝜌 has many more potential applications than the divisorial Tate conjecture (e.g., CM lifting and the Tate conjecture for self-correspondences, as shown in [29]).

On the other hand, our main purpose is to deal with the situation when the smoothness of 𝜌 cannot be hoped for. In particular, Theorem B implies that the characteristic 𝑝 counterparts of the surfaces in Moonen’s list [49, Theorem 9.4] satisfy the Tate conjecture for p ≫ 0 , and we expect that our methods for the refinements, Theorems A and C, can be adapted to make Theorem B effective for most classes of these surfaces.

Organization of paper

In Section 2, we review and mildly extend Moonen’s results in [49] on the motives of fibers of ♡ -families. In particular, we recap the norm functors used in [49]. In Section 3, we discuss the moduli interpretations of Shimura varieties of abelian type over their reflex fields, as documented in [63], and recap the integral models of orthogonal Shimura varieties from [41]. In Section 4, we construct the period morphisms for ♡ -families in characteristic 0, using results from Sections 2 and 3. In Section 5, we prove Theorem B after giving a more effective version (see Proposition 5.3.3). In Section 6, we set up some basic tools to analyze deformation of curves on parameter spaces, including applications of the Baire category theorem. Finally, in Sections 7 and 8, we use tools from Section 6 as well as Proposition 5.3.3 to prove Theorems A and C respectively. In particular, in Section 7, we study the geometry of natural parameter spaces of elliptic surfaces, which we hope to be of independent interest.

1.5

Finally, we introduce some notation and conventions.

Let f : X → S be a morphism between schemes. If T → S is another morphism, denote by X T the base change X × S T and by X ⁢ ( T ) the set of morphisms T → X as 𝑆-schemes. By a geometric fiber of 𝑋, we mean X s for some geometric point s → S . If 𝑥 is a point (resp. geometric point) on a scheme 𝑋, we write k ⁢ ( x ) for its residue field (resp. field of definition). By a variety over a field 𝑘, we mean a scheme which is reduced, separated, and of finite type over 𝑘.
The letters 𝑝 and ℓ will always denote some prime numbers and ℓ ≠ p unless otherwise noted. We write Z ̂ for the profinite completion of ℤ and Z ̂ p for its prime-to-𝑝 part. Define A f : = Z ̂ ⊗ Q and A f p : = Z ̂ p ⊗ Q . If 𝑘 is a perfect field of characteristic 𝑝, we write W ⁢ ( k ) for its ring of Witt vectors.
For a field 𝑘 of characteristic ≠ 2 , we consider a quadratic form 𝑞 on a finite-dimensional 𝑘-vector space 𝑉 simultaneously a symmetric bilinear pairing ⟨ − , − ⟩ such that q ⁢ ( v ) = ⟨ v , v ⟩ for every v ∈ V .
For a finite free module 𝑀 over a ring 𝑅, we write M ⊗ for the direct sum of all the 𝑅-modules which can be formed from 𝑀 by taking duals, tensor products, symmetric powers, and exterior powers. We also use this notation for sheaves of modules on some Grothendieck topology whenever it makes sense.
We use the following abbreviations: VHS for “variations of Hodge structures” and ODP for “ordinary double point”. Unless otherwise noted, the local system in a VHS is a local system of ℚ-vector spaces. Moreover, we always assume that the VHS is pure, i.e., it is a direct sum of those pure of some weight (or the weight filtration is split).

2 Preliminaries

2.1 Motives and norm functors

2.1.1

Let 𝑘 be a field of characteristic 0. We denote by Mot AH ⁢ ( k ) the neutral ℚ-linear Tannakian category of motives over 𝑘 with morphisms defined by absolute Hodge correspondences (cf. [51, §2], where it is denoted by M k ). We have the Tate objects 1 ⁢ ( n ) for every n ∈ Z in this category. For any object M ∈ Mot AH ⁢ ( k ) , we write M ⁢ ( n ) for M ⊗ 1 ⁢ ( n ) , and by an absolute Hodge cycle on 𝑀, we mean a morphism 1 → M .

Following [40, §2], we denote by ω ℓ the ℓ-adic realization functor which sends Mot AH ⁢ ( k ) to the category of finite-dimensional Q ℓ -vector spaces with an action of Gal k : = Gal ( k ̄ / k ) , where k ̄ is some chosen algebraic closure of 𝑘. Putting ω ℓ ’s together, we obtain ω ét , which takes values in the category of finite free A f -modules with a Gal k -action. Let ω dR denote the de Rham realization functor, which takes values in the category of filtered 𝑘-vector spaces. If 𝑘 is a subfield of ℂ, we additionally consider the Betti realization ω B after base change to ℂ (resp. the Hodge realization ω Hdg ) which takes values in the category of ℚ-vector spaces (resp. Hodge structures). For a smooth projective variety 𝑋 over 𝑘, h i ⁢ ( X ) denotes the object such that ω ? ⁢ ( h i ⁢ ( X ) ) is the 𝑖th ? -cohomology of 𝑋, for ? = B , ℓ , dR whenever applicable.

Let Mot Ab ⁢ ( k ) ⊆ Mot AH ⁢ ( k ) be the full Tannakian subcategory generated by the Artin motives and the motives attached to abelian varieties. We will repeatedly make use of the following fact ([16, Chapter I]; cf. [40, Theorem 2.3]).

Theorem 2.1.2

The functor ω Hdg is fully faithful when restricted to Mot Ab ⁢ ( C ) . In particular, for every M ∈ Mot Ab ⁢ ( C ) , every element s ∈ ω B ⁢ ( M ) ∩ Fil 0 ⁢ ω dR ⁢ ( M ) is given by an absolute Hodge cycle.

We often refer to objects in Mot Ab ⁢ ( k ) as abelian motives.

2.1.3

We will often consider the automorphism − ⊗ σ C on Mot AH ⁢ ( C ) defined by an element σ ∈ Aut ⁢ ( C ) (cf. [16, §II 6.7]; see also [40, Proposition 2.2]). For M ∈ Mot AH ⁢ ( C ) , we write M σ for M ⊗ σ C . Base change properties of étale (resp. de Rham) cohomology give us an A f -linear (resp. 𝜎-linear) canonical isomorphism

ω ét ⁢ ( M ) ≅ bc ω ét ⁢ ( M σ ) ( resp. ⁢ ω dR ⁢ ( M ) ≅ bc ω dR ⁢ ( M σ ) ) .

Here the subscript “bc” is short for “base change”. For an absolute Hodge class s ∈ ω B ⁢ ( M ) , we write s σ for the class in ω B ⁢ ( M σ ) which has the same étale and de Rham realizations as 𝑠 under the “ ≅ bc ” isomorphisms.

Finally, we remark that, for M ∈ Mot AH ⁢ ( C ) , it makes sense to say whether a tensor s ∈ ω B ⁢ ( M ) ⊗ (see Section 1.5 (d)) is absolute Hodge, because 𝑠 has to lie in ω B of a finite direct sum of tensorial constructions on 𝑀. And when 𝑠 is absolute Hodge, we may form s σ ∈ ω B ⁢ ( M σ ) ⊗ for σ ∈ Aut ⁢ ( C ) , extending the notation in the previous paragraph.

2.1.4

Next, we recall the basics of norm functors. The reader may refer to [49, §3] for more details. Let 𝑘 be a field of characteristic 0 and let 𝐸 be a finite étale 𝑘-algebra. Let 𝒞 be any Tannakian 𝑘-linear category and Mod E ⁢ ( C ) the category of 𝐸-modules in 𝒞. For any object M ∈ Mod E ⁢ ( C ) , we write M ( k ) for the underlying object in 𝒞 when we forget the 𝐸-linear structure. In [20], Ferrand gave a general construction of a norm functor Nm E / k : Mod E ⁢ ( C ) → C , which was summarized in [49, §3.6].

We first study the case when 𝒞 is the category of 𝑘-modules Mod k . For any M ∈ Mod E , there is a functorial polynomial map ν M : M → Nm E / k ⁢ ( M ) such that

ν M ⁢ ( e ⁢ m ) = Norm E / k ⁢ ( e ) ⁢ ν M ⁢ ( m ) for any ⁢ e ∈ E ⁢ and ⁢ m ∈ M .

The norm functor Nm E / k is a ⊗-functor and is non-additive (unless E = k ). However, for any M 1 , M 2 ∈ Mod E , there is an identification

Nm E / k ⁢ ( Hom E ⁢ ( M 1 , M 2 ) ) = Hom k ⁢ ( Nm E / k ⁢ ( M 1 ) , Nm E / k ⁢ ( M 2 ) ) .

For any V ∈ Mod E , there is a natural map

η : Res E / k ⁡ GL ⁢ ( V ) → GL ⁢ ( Nm E / k ⁢ ( V ) )

which sends an 𝐸-linear automorphism 𝑓 to Nm E / k ⁢ ( f ) . Denote the torus Res E / k ⁡ G m , E by T E / k and the kernel of the norm map T E / k → T k by T E / Q 1 . When G m , E is viewed as the diagonal torus of GL ⁢ ( V ) , so that T E / k 1 is a subgroup of Res E / k ⁡ GL ⁢ ( V ) , we have ker ⁡ ( η ) = T E / k 1 .

Notation 2.1.5

Take k = Q and let 𝐸 be a totally real field. Let 𝒱 be an 𝐸-vector space equipped with a quadratic form ϕ ̃ : V → E . We often drop ϕ ̃ from the notation when it is assumed.

Write V ( Q ) for the underlying ℚ-vector space of 𝒱, equipped with a quadratic form ϕ : V ( Q ) → Q given by tr E / Q ∘ ϕ ̃ .
Write G ⁢ ( V ) for Res E / Q ⁡ SO ⁢ ( V ) , Z ⁢ ( V ) for T E / Q 1 ∩ G ⁢ ( V ) , and H ⁢ ( V ) for G ⁢ ( V ) / Z ⁢ ( V ) , or simply 𝒢, 𝒵, and ℋ when 𝒱 is understood.
Denote by Cl + ⁢ ( V ) the even Clifford algebra of 𝒱 over 𝐸 and by Cl E / Q + ⁢ ( V ) its norm Nm E / Q ⁢ Cl + ⁢ ( V ) .
Let ℰ denote the 1-dimensional quadratic form over 𝐸 given by equipping 𝐸 with the form ϕ ̃ ⁢ ( α ) = α 2 .

Recall our convention in Section 1.5 (c). The association V ↦ V ( Q ) in (a) above defines an equivalence of categories between quadratic forms over 𝐸 and quadratic forms over ℚ with a self-adjoint 𝐸-action [35, Chapter 1, Theorem 7.4.1]. We call V ( Q ) the transfer of 𝒱, and 𝒱 the 𝐸-bilinear lift of V ( Q ) .

2.1.6

Let us recall some formalism about motives for readers’ convenience. Let 𝑘 be a subfield of ℂ. For any object m ∈ Mot ⁢ ( k ) or Mot AH ⁢ ( k ) , let G mot ⁢ ( m ) be the motivic Galois group of 𝔪, which is defined in [2, §4.6]. Here we use Betti cohomology as the reference Weil cohomology theory, through which the Tannakian subcategory ⟨ m ⟩ ⊆ Mot AH ⁢ ( k ) generated by 𝔪 is naturally equivalent to the category of G mot ⁢ ( m ) -representations over ℚ. In particular, 𝔪 itself corresponds to a representation G mot ⁢ ( m ) → GL ⁢ ( ω B ⁢ ( m ) ) such that every vector or subspace of ω B ⁢ ( m ) ⊗ invariant under G mot ⁢ ( m ) is defined by a submotive of m ⊗ . The reader may check out [48, §3.1] for an exposition of these notions.

Lemma 2.1.7

Let 𝐸 be a totally real field, and let w ∈ Mot AH ⁢ ( C ) be a motive equipped with an 𝐸-action and a symmetric 𝐸-bilinear form ϕ ̃ : w ⊗ E w → 1 E . If dim E ω B ⁢ ( w ) is odd, then the motive N ( w ) : = Nm E / Q ( w ) ⊗ Q det ( w ( Q ) ) is (noncanonically) isomorphic to a submotive of Cl E / Q + ⁢ ( w , ϕ ̃ ) .

The meaning of Cl E / Q + ⁢ ( w , ϕ ̃ ) is explained in the proof.

Proof

This follows from the content in [49, §5.4]. We give a sketch so that the reader can easily check the details from [49, §5.4]. Set W : = ω B ( w ) and let 𝒲 be the 𝐸-bilinear lift of 𝑊. Since the 𝐸-action and the pairing ϕ ̃ are motivic, the representation G mot ⁢ ( w ) → GL ⁢ ( W ) defined by 𝔴 takes values in O E / Q ⁢ ( W , ϕ ̃ ) . Then

Cl E / Q + ⁢ ( w , ϕ ̃ ) ∈ ⟨ w ⟩

is the motive defined by the adjoint representation of G mot ⁢ ( w ) on Cl E / Q + ⁢ ( W , ϕ ̃ ) .

Let Σ be the set of embeddings σ : E ↪ C , let 𝔖 be the symmetric group of Σ, and let 𝑄 be the set of 𝔖-orbits of N Σ . Then, under the assumption that dim E W is odd, there is an ascending filtration Fil ∙ on Cl E / Q + ⁢ ( w , ϕ ̃ ) indexed by 𝑄 such that, for some q 1 , q 2 ∈ Q , Fil q 1 / Fil q 2 ≅ N ⁢ ( w ) . Therefore, N ⁢ ( w ) is a subquotient of Cl E / Q + ⁢ ( w , ϕ ̃ ) . As Mot AH ⁢ ( C ) is semisimple, N ⁢ ( w ) is in fact (noncanonically) a sub-object. ∎

2.2 Motives of varieties with h 2 , 0 = 1

Definition 2.2.1

A polarized Hodge structure 𝑉 of weight 0 is said to be of K3-type if dim V C ( − 1 , 1 ) = dim V C ( 1 , − 1 ) = 1 , and V C i , j = 0 when | i − j | > 2 . The transcendental part T ⁢ ( V ) of 𝑉 is the orthogonal complement of V ( 0 , 0 ) : = V ∩ V C ( 0 , 0 ) .

We recall the following fundamental result of Zarhin.

Theorem 2.2.2

Theorem 2.2.2 ([65, §2])

Let 𝑉 be a Hodge structure of K3-type such that V ( 0 , 0 ) = 0 and let 𝜙 be its polarization form. Then the endomorphism algebra E : = End Hdg V is either a totally real field or a CM field, and the adjoint map e ↦ e ̄ defined by ϕ ⁢ ( e ⁢ x , y ) = ϕ ⁢ ( x , e ̄ ⁢ y ) is the identity map when 𝐸 is totally real and is complex conjugation when 𝐸 is CM.

To discuss motives in families, we first give a definition.

Definition 2.2.3

Let 𝑆 be a connected smooth ℂ-variety. For every ℚ-local system V B over 𝑆 and b ∈ S ⁢ ( C ) , we write Mon ⁢ ( V B , b ) for the Zariski closure of the image of π 1 ⁢ ( S , b ) in GL ⁢ ( V B , b ) , and Mon ∘ ⁢ ( V B , b ) for its identity component. When V = ( V B , V dR ) is a polarizable VHS^[4] over 𝑆, we say that 𝖵 has maximal monodromy if Mon ∘ ⁢ ( V B , b ) is equal to the derived group of the Mumford–Tate group MT ⁢ ( V b ) , where 𝑏 is any Hodge-generic point.

For the terminology “Hodge-generic points”, see for example [48, 31]. Note that [1, §5] says that, in the above notation, Mon ∘ ⁢ ( V B , b ) is always a normal subgroup of the derived group of MT ⁢ ( V b ) (see also [52, Theorem 16]).

2.2.4

For the rest of Section 2.2, let 𝑆 be a connected smooth ℂ-variety and f : X → S a ♡ -family of relative dimension 𝑑. Let V = ( V B , V dR ) be the VHS on 𝑆 defined by R 2 ⁢ f ∗ ⁢ Q ⁢ ( 1 ) . Note that, by the definition of a ♡ -family, 𝖵 satisfies condition (P) in [49, Proposition 6.4]. Let 𝝃 be a relatively ample line bundle on X / S , which defines a symmetric bilinear pairing ⟨ − , − ⟩ on 𝖵 such that ⟨ α , β ⟩ = α ∪ β ∪ c 1 ⁢ ( ξ ) d − 2 for local sections α , β . Choose a Hodge-generic base point b ∈ S ⁢ ( C ) .

Suppose for a moment that Mon ⁢ ( V B , b ) is connected. Then, for ρ : = dim NS ( X b ) Q , 𝖵 admits an orthogonal decomposition 1 S ⊕ ρ ⊕ P , where 𝖯 is a VHS polarized by the pairing induced by 𝝃 and 1 S is the unit VHS. Note that the Hodge structure P b = ( P B , b , P dR , b ) , together with its polarization, is of K3-type and satisfies the hypothesis of Theorem 2.2.2. Let 𝐸 be the endomorphism field of P b . As Mon ⁢ ( V B , b ) = Mon ∘ ⁢ ( V B , b ) ⊆ MT ⁢ ( V b ) = MT ⁢ ( P b ) and MT ⁢ ( P b ) commutes with 𝐸, the 𝐸-action on P B , b commutes with π 1 ⁢ ( S , b ) and hence extends to an action on 𝖯 (see, e.g., [52, Corollary 12]). If 𝖵 (or equivalently 𝖯) has maximal monodromy, we say that ( X / S , ξ ) is in case (R+) if 𝐸 is totally real and (CM) if 𝐸 is CM; we further divide (R+) into case (R1) for dim E P B , b odd and case (R2) for dim E P B , b even. We say we are in case (R2^′) if 𝖵 has non-maximal monodromy, which can only happen when 𝐸 is totally real and dim E P B , b = 4 . See [49, Proposition 6.4 (iii)] and its proof.

If Mon ⁢ ( V B , b ) is not connected, we say that ( X / S , ξ ) is in case ? for ? = (R1) , (R2), (CM) or (R2^′) if it is in case ? up to replacing 𝑆 by a connected étale cover S ′ and 𝑏 by a lift such that Mon ⁢ ( V B , b ) becomes connected. The definition is clearly independent of these choices.

Proposition 2.2.5

Suppose 𝖵 has non-maximal monodromy, or equivalently ( X / S , ξ ) belongs to case (R2^′). Then, for a general s ∈ S and X : = X s , the Kodaira–Spencer map ∇ s : T s ⁢ S → Hom ⁢ ( H 1 ⁢ ( Ω X 1 ) , H 2 ⁢ ( O X ) ) has rank 1.

Proof

We may assume that Mon ⁢ ( V B , b ) is connected, so that there is a decomposition 1 S ⊕ ρ ⊕ P as above. The rank of ∇ s achieves its maximum on an open dense U ⊆ S . Choose some s ∈ U . By [61, Theorem 3.5] (cf. [49, Proposition 6.4]), for some point 𝑧 in a small analytic neighborhood of 𝑠, P B , z ( 0 , 0 ) contains a nonzero class 𝜁. Let Z ′ ⊆ S be the irreducible component of the Noether–Lefschetz loci defined by ( z , ζ ) and let 𝑍 be its smooth locus. Up to replacing 𝑧 by a different point on Z ′ , we may assume that 𝑧 lies in 𝑍, is Hodge-generic for the VHS P | Z , and rank ⁢ ∇ s = rank ⁢ ∇ z . Let 𝑊 be the Hodge structure of K3-type defined by the fiber of 𝖯 at 𝑧 and let 𝑇 be its transcendental part. Then dim E T < dim E W = 4 . Set F : = End Hdg T , which contains 𝐸.

By assumption on a ♡ -family, we have rank ⁢ ∇ s > 0 . Suppose by way of contradiction that rank ⁢ ∇ s > 1 . Then, as Z ⊆ S has codimension 1 (cf. [61, Lemma 3.1]), ∇ z does not vanish on T z ⁢ Z . This implies that Mon ∘ ⁢ ( P B | Z , z ) ≠ 1 and dim E T = 3 by [49, Proposition 6.4] and its proof. Indeed, if 𝐹 is CM, then dim F T ≥ 2 , so that dim E T ≥ 4 , which is impossible. Therefore, 𝐹 is totally real and dim F T ≥ 3 . This forces E = F and dim E T = 3 . Now let 𝒯 be the 𝐸-bilinear lift of 𝑇. Then, by [65, Theorem 2.2.1], MT ⁢ ( T ) = Res E / Q ⁡ SO ⁢ ( T ) . As this group is simple, Mon ∘ ⁢ ( P B | Z , z ) ≅ Res E / Q ⁡ SO ⁢ ( T ) . On the other hand, [49, §8.1] tells us that Mon ∘ ⁢ ( P B , s ) ≅ Res E / Q ⁡ L for some 𝐸-form 𝐿 of SL 2 . However, as SL 2 and SO ⁢ ( 3 ) are not even isomorphic over ℂ, Res E / Q ⁡ L cannot have a subgroup isomorphic to Res E / Q ⁡ SO ⁢ ( T ) , which contradicts the fact that parallel transport (noncanonically) sends Mon ∘ ⁢ ( P B | Z , z ) into Mon ∘ ⁢ ( P B , s ) . ∎

2.2.6

The following statements are the key inputs that we will use from Moonen’s paper [49]. A little adaptation we make is that we will uniformly use the category of motives Mot AH ⁢ ( C ) with absolute Hodge cycles, whereas Moonen used André’s category with motivated cycles [2]. This adaptation makes a difference only for Theorem 2.2.7 (d) below. We use Mot ⁢ ( C ) to denote André’s category (for base field ℂ) when explaining this difference, but otherwise all motives are considered in Mot AH ⁢ ( C ) . Note that motivated cycles are automatically absolute Hodge, so that Mot ⁢ ( C ) is a subcategory of Mot AH ⁢ ( C ) .

Recall our notation in Sections 2.1.4 and 2.2.4 for the theorem below.

Theorem 2.2.7

Assume that Mon ⁢ ( V B , b ) is connected and let 𝖯 and 𝐸 be as in Section 2.2.4. Let s ∈ S ⁢ ( C ) be any point and write p s ∈ Mot AH ⁢ ( C ) for the submotive of h 2 ⁢ ( X s ) ⁢ ( 1 ) such that ω Hdg ⁢ ( p s ) = P s . Then the action of 𝐸 on P s is absolute Hodge, i.e., induced by an action of 𝐸 on p s . Moreover,

in case (R1), Nm E / Q ⁢ ( p s ) ⊗ Q det ( p s , ( Q ) ) is an object of Mot Ab ⁢ ( C ) .
In case (R2), for 1 E : = 1 ⊗ E and p s ♯ : = p s ⊕ 1 E , Nm E / Q ⁢ ( p s ♯ ) ⊗ Q det ( p s , ( Q ) ♯ ) is an object of Mot Ab ⁢ ( C ) .
In case (R2^′), Nm E / Q ⁢ ( p s ) is an object of Mot Ab ⁢ ( C ) .
In case (CM), p s is an object of Mot Ab ⁢ ( C ) .

Proof

The statement that the action of 𝐸 on P s is absolute Hodge is implied by [49, Proposition 6.6]. Note that our p s is Moonen’s V s , and our 𝖯 is Moonen’s 𝕍. Below, we view p s as an object of Mot AH ⁢ ( C ) ( E ) . In this proof, End ¯ (or Hom ¯ ) always mean internal End (or Hom ) in the category Mot AH ⁢ ( C ) .

(a) We first treat the case (R+). Let 𝒱 be the 𝐸-bilinear lift of the quadratic form given by P B , b with its self-dual 𝐸-action. By [49, §§6.9, 6.10], there is a family of abelian schemes A → S with multiplication by D : = Cl E / Q + ( V ) (see Notation 2.1.5) such that there is an isomorphism

(2.1) Cl E / Q + ⁢ ( p s ) ⁢ → ∼ ⁢ End ¯ D ⁢ ( h 1 ⁢ ( A s ) )

of algebra objects in Mot AH ⁢ ( C ) . In case (R1), dim E V is odd, and by Lemma 2.1.7,

Nm E / Q ⁢ ( p s ) ⊗ det ( p s , ( Q ) )

is noncanonically a submotive of Cl E / Q + ⁢ ( p s ) , and hence is an object of Mot Ab ⁢ ( C ) .

(b) In case (R2), (2.1) still holds, so that Cl E / Q + ⁢ ( p s ) is still an object of Mot Ab ⁢ ( C ) , but a further trick is needed to recover (a variant of) Nm E / Q ⁢ ( p s ) from Cl E / Q + ⁢ ( p s ) . Recall ℰ defined in Notation 2.1.5 (d). As in [49, §§6.11, 6.12], we consider the VHS

P ♯ : = P ⊕ ( 1 S ⊗ E ( Q ) ) ,

where 1 S stands for the unit VHS on 𝑆. Let V ♯ : = V ⊕ E and D ♯ : = Cl E / Q + ( V ♯ ) . Then, by [49, §§6.9, 6.10] again, there is an abelian scheme A ♯ → S with multiplication by D ♯ such that

(2.2) Cl E / Q + ⁢ ( P ♯ ) ≅ End ¯ D ♯ ⁢ ( H ) ,

where 𝖧 is the VHS given by the first relative cohomology of A ♯ . In [49, §§6.9, 6.10], it is shown that the fiber of the isomorphism (2.2) at every ℂ-point on 𝑆 is induced by an absolute Hodge cycle. Here is a summary of the argument in our notation. Choose a point s 0 ∈ S ⁢ ( C ) such that P s 0 ( 0 , 0 ) ≠ 0 . By the last paragraph of [49, §6.12], there is an isomorphism

Cl E / Q + ⁢ ( p s 0 ♯ ) ≅ Cl E / Q + ⁢ ( p s 0 ) ⊕ 2 [ E : Q ]

of objects in Mot AH ⁢ ( C ) . As Cl E / Q + ⁢ ( p s 0 ) is an object of Mot Ab ⁢ ( C ) , so is Cl E / Q + ⁢ ( p s 0 ♯ ) . Therefore, by Theorem 2.1.2, the isomorphism (2.2) is absolute Hodge at s 0 , and hence so at every other 𝑠 by Deligne’s Principle B. This implies Cl E / Q + ⁢ ( p s ♯ ) ∈ Mot Ab ⁢ ( C ) , so that (b) follows from Lemma 2.1.7 again.

(d) In case (CM), [49, §7.4] tells us that there exist a motive 𝔲 (denoted by 𝑼 therein), an abelian variety 𝐴 over ℂ, and an abelian scheme B → S , all equipped with multiplication by 𝐸, such that there is an isomorphism

p s ⊗ E u → ∼ h s : = Hom ¯ E ( h 1 ( A ) , h 1 ( B s ) ) .

This implies that p s ⊗ u ∈ Mot Ab ⁢ ( C ) . In order to show p s ∈ Mot Ab ⁢ ( C ) , it suffices to argue that u ≅ 1 E in Mot AH ⁢ ( C ) .

In [49, §7.4], 𝔲 is in fact constructed as an object Mot ⁢ ( C ) . Moonen remarked that, conjecturally, there should be an isomorphism u ≅ 1 E in Mot ⁢ ( C ) and proved that 𝔲 indeed has trivial Hodge and ℓ-adic realizations. We note that his argument in [49, Lemma 7.5] in fact implies that u ≅ 1 E in Mot AH ⁢ ( C ) , i.e., ω B ⁢ ( u ) is spanned by absolute Hodge classes: the idea is to take advantage of the fact that 𝔲 is independent of 𝑠, and that, for some s 0 ∈ S , the algebraic part ω B ⁢ ( p s 0 ) ( 0 , 0 ) of ω B ⁢ ( p s 0 ) is nonempty. Since every class in ω B ⁢ ( u ) is of type ( 0 , 0 ) , we have ω B ⁢ ( p s 0 ) ( 0 , 0 ) ⊗ E ω B ⁢ ( u ) = ω B ⁢ ( h s 0 ) ( 0 , 0 ) . By the Lefschetz ( 1 , 1 ) -theorem, every class in ω B ⁢ ( p s 0 ) ( 0 , 0 ) comes from a line bundle on X s and hence is absolute Hodge. As classes in ω B ⁢ ( h s 0 ) ( 0 , 0 ) are also absolute Hodge, we may now conclude by Proposition 2.2.8 below. ∎

Proposition 2.2.8

Let 𝐸 be a number field and let m , n be objects of Mot AH ⁢ ( C ) with 𝐸-action. Let h : = m ⊗ E n . Let m ∈ ω B ⁢ ( m ) , n ∈ ω B ⁢ ( n ) be nonzero Hodge cycles and define h ∈ ω B ⁢ ( h ) to be m ⊗ E n . If ℎ and 𝑚 are both absolute Hodge, then so is 𝑛.

Proof

Recall notation in Section 2.1.3. We need to show that, for every σ ∈ Aut ⁢ ( C ) , the image n ′ of n ⊗ 1 under the canonical isomorphisms

ω B ⁢ ( n ) ⊗ ( C ⊗ A f ) ≅ ω dR ⁢ ( n ) × ω A f ⁢ ( n ) ≅ bc ω dR ⁢ ( n σ ) × ω A f ⁢ ( n σ ) ≅ ω B ⁢ ( n σ ) ⊗ ( C ⊗ A f ) .

is contained in ω B ⁢ ( n σ ) (i.e., is of the form n σ ⊗ 1 for some n σ ∈ ω B ⁢ ( n σ ) ). Since 𝑚 and ℎ are absolute Hodge, we know that m σ ∈ ω B ⁢ ( m σ ) and h σ = m σ ⊗ n ′ ∈ ω B ⁢ ( h σ ) . Then apply Lemma 2.2.9 with k = Q , R = A f × C , M = ω B ⁢ ( m σ ) , and N = ω B ⁢ ( n σ ) to deduce that n ′ ∈ ω B ⁢ ( n σ ) . ∎

Lemma 2.2.9

Let E / k be a field extension, 𝑅 any 𝑘-algebra and M , N finite-dimensional 𝐸-vector spaces. Let H : = M ⊗ E N and let h ∈ H ⊗ k R be a nonzero element of the form m ⊗ n under the canonical isomorphism

H ⊗ k R ≅ ( M ⊗ k R ) ⊗ E ⊗ k R ( N ⊗ k R ) ,

where m ∈ M ⊗ k R and n ∈ N ⊗ k R . If h ∈ H and m ∈ M , then n ∈ N .

Proof

Let α = dim E M and β = dim E N . By choosing bases of 𝑀 and 𝑁, we may assume M = E α and N = E β . Identify 𝐻 with the space of ( α × β ) -matrices over 𝐸 and ℎ with m ⋅ n T . We denote by ( m i ) , ( n j ) , ( h i , j ) the respective E ⊗ k R -coordinates of 𝑚, 𝑛, and ℎ and fix 𝑖 such that m i ≠ 0 . Then, for every 𝑗, h i , j = m i ⋅ n j and hence n j = m i − 1 ⁢ h i , j ∈ E because m i , h i , j ∈ E ⊆ E ⊗ k R by assumption. ∎

3 Moduli interpretations of Shimura varieties

In this section, we first recall the moduli description for the canonical models of some Shimura varieties of abelian type from [63]. Then we give some preliminary results on those of orthogonal type over totally real fields and review their integral models when the reflex field is ℚ.

3.1 Systems of realizations

Definition 3.1.1

Let 𝑘 be a subfield of ℂ and let 𝑆 be a smooth 𝑘-variety. By a system of realizations, we mean a tuple V = ( V B , V dR , V ét , i dR , i ét ) where

V B is a ℚ-local system over S C : = S ⊗ k C ;
V dR is a filtered flat vector bundle over 𝑆;
V ét is an étale local system of A f -coefficients over 𝑆;
i dR : ( V B ⊗ O S C an , id ⊗ d ) ⁢ → ∼ ⁢ ( V dR | S C ) an is an isomorphism of flat (holomorphic) vector bundles such that ( V B , V dR | S C ) is a polarizable VHS;
i ét : V B ⊗ A f ⁢ → ∼ ⁢ V ét | S C is an isomorphism between (the pro-étale sheaf associated to) V B ⊗ A f and V ét | S C .

We may often omit ( i dR , i ét ) in the notation and write

V B ⊗ A f = V ét | S C and V B ⊗ O S C an = V dR | S C

a little abusively. Let R ⁢ ( S ) denote the category of systems of realizations over 𝑆, with morphisms defined in the obvious way. Then R ⁢ ( S ) is naturally a Tannakian category. Let 1 S be its unit object, and for each n ∈ Z , let 1 S ⁢ ( n ) be the Tate object. For every V ∈ R ⁢ ( S ) , set V ( n ) : = V ⊗ 1 S ( n ) . Note that if k = C , then R ⁢ ( S ) is naturally identified with the category of polarizable VHS over 𝑆. We write H 0 ⁢ ( V ) for Hom ⁢ ( 1 , V ) .

Our definition is a little different from [21, §6.1] because we fixed an embedding k ↪ C , but see [63, (3.1.3)] for a comparison. Also, recall our convention in Section 1.5 (e) for VHS. We remind the reader that, since 𝑆 is defined over k ⊆ C , when we write s ∈ S ⁢ ( C ) , we mean a 𝑘-linear morphism Spec ⁡ ( C ) → S , which we view simultaneously as a closed point on S C = S ⊗ k C .

Next, we recall how to define a 𝖪-level structure in this context. Interestingly, this definition can be leveraged to define additional structures on 𝖵, as Example 3.1.3 below shows.

Definition 3.1.2

Let 𝐺 be a reductive group over ℚ, let V ∈ Rep ⁢ ( G ) be a finite-dimensional 𝐺-representation and let K ⊆ G ⁢ ( A f ) be a compact open subgroup. For any subfield k ⊆ C , smooth 𝑘-variety 𝑆, and any system of realizations V = ( V B , V dR , V ét ) on 𝑆, we make the following definitions.

A ( G , V , K ) -level structure on V ét , or simply a 𝖪-level structure, is a global section [ η ] of the quotient sheaf K \ Isom ¯ ⁢ ( V ⊗ A f , V ét ) . Here Isom ¯ ⁢ ( V ⊗ A f , V ét ) denotes the pro-étale sheaf consisting of isomorphisms V ⊗ A f ⁢ → ∼ ⁢ V ét , and 𝖪 acts by pre-composition through its image in GL ⁢ ( V ⊗ A f ) . For each geometric point s → S , write the 𝖪-orbit of isomorphisms V ⊗ A f ⁢ → ∼ ⁢ V ét , s determined by [ η ] as [ η ] s .
If [ η ] is a ( G , V , K ) -level structure on V ét , every element v ∈ I : = ( V ⊗ ) G gives rise to a global section of V ét , which we denote by η ⁢ ( v ) ét , such that, for every geometric point s → S , every representative of [ η ] s sends v ⊗ 1 to η ⁢ ( v ) ét , s . We say that [ η ] is 𝖵-rational if (i) for all v ∈ I , there exists a global section v = ( v B , v dR , v ét ) ∈ H 0 ⁢ ( V ⊗ ) such that v ét = η ⁢ ( v ) ét , and (ii) for every s ∈ S ⁢ ( C ) , ( V B , s , { v B , s } v ∈ I ) ≅ ( V , { v } v ∈ I ) .
Let Ω be a G ⁢ ( R ) -conjugacy class of morphisms S → G R . When 𝑉 is faithful, we say that a 𝖵-rational 𝖪-level structure is of type Ω if, for every s ∈ S ⁢ ( C ) , under some (and hence every) isomorphism ( V B , s , { v B , s } v ∈ I ) ≅ ( V , { v } v ∈ I ) , the Hodge structure on V B , s is defined by an element of Ω.
If there is a morphism G ′ → G from another reductive subgroup G ′ , and K ′ ⊆ G ′ ⁢ ( A f ) is a compact open subgroup whose image is contained in 𝖪, then we say that a K ′ -level structure [ η ′ ] on V ét refines [ η ] if [ η ] is induced by [ η ′ ] via the natural (forgetful) map K ′ \ Isom ¯ ⁢ ( V ⊗ A f , V ét ) → K \ Isom ¯ ⁢ ( V ⊗ A f , V ét ) .

Definitions (b) and (c) above are used to simplify the formalism of the moduli interpretation of Shimura varieties. To explain this, we give a PEL-type example.

Example 3.1.3

Let ( B , ∗ , ( V , ψ ) ) be a simple PEL-datum (i.e., 𝐵 is a simple ℚ-algebra with positive involution ∗ of type 𝐴 or 𝐶 and ( V , ψ ) is a symplectic 𝐵-module). We denote by G : = GSp B ( V ) the ℚ-group of 𝐵-linear similitudes and fix an open compact subgroup K ⊂ G ⁢ ( A f ) . Then there exists a unique G ⁢ ( R ) -conjugacy class Ω such that ( G , Ω ) is a Shimura datum and Sh K ⁢ ( G , Ω ) ⁢ ( C ) is in canonical bijection with the set ℐ of isomorphism classes of tuples ( V , λ , i , [ η ] ) , where

𝖵 is a ℚ-Hodge structure ( V B , V dR ) of type { ( 1 , 0 ) , ( 0 , 1 ) } ,
i : B ↪ End ⁡ ( V ) is an algebra morphism,
𝜆 is a Q × -equivalence class of 𝐵-linear symplectic pairing of 𝖵 and
[ η ] is a 𝖪-orbit of a 𝐵-linear similitude V ⊗ A f → V ét

such that ( ♠ ) there exists a 𝐵-linear similitude V ≅ V B through which the Hodge structure on 𝖵 is defined by an element of Ω (see, e.g., [47, Proposition 8.14, Theorem 8.17]). Here we implicitly applied the equivalence A ↦ H 1 ⁢ ( A , Q ) between the category of complex abelian varieties up to isogeny and that of polarizable Hodge structures of type { ( 1 , 0 ) , ( 0 , 1 ) } .

Note that an object of R ( C ) : = R ( Spec ( C ) ) is nothing but a polarizable Hodge structure. We can more concisely define ℐ as the isomorphism classes of pairs ( V , [ η ] ) , where

V ∈ R ⁢ ( C ) and
[ η ] is a 𝖵-rational ( G , V , K ) -level structure on V ét of type Ω.

Indeed, if we view each b ∈ B as a tensor in End Q ⁡ ( V ) = V ∨ ⊗ V and c : = Q × ⋅ ψ as a tensor in V 2 ⊗ ( V ∨ ) 2 (cf. [32, Example 2.1.6]), we have i ⁢ ( b ) ⊗ A f = η ⁢ ( b ) ét and λ ⊗ A f = η ⁢ ( c ) ét in the notation of Definition 3.1.2 (b). In particular, the datum of 𝑖 and 𝜆 is remembered by the condition that η ⁢ ( b ) ét and η ⁢ ( c ) ét come from global sections (i.e., elements) of V B ⊗ of Hodge type ( 0 , 0 ) . Since 𝐺 is the stabilizer of 𝐵 and 𝑐 in GL ⁢ ( V ) , we may equivalently say that this is true for all v ∈ ( V ⊗ ) G . Therefore, the existence of 𝑖 and 𝜆 such that there exists a 𝐵-linear similitude V ≅ V B is equivalent to [ η ] being 𝖵-rational. Then condition ( ♠ ) can be simply stated as “ [ η ] is of type Ω” in the sense of Definition 3.1.2 (c).

Now we introduce some notation to keep track of Galois descent data in a system of realizations.

Notation 3.1.4

Let 𝑆 be a smooth variety over a subfield 𝑘 of ℂ. Let s ∈ S ⁢ ( C ) and σ ∈ Aut ⁢ ( C / k ) be any elements. Denote by σ ⁢ ( s ) ∈ S ⁢ ( C ) the point given by pre-composing the 𝑘-linear morphism s : Spec ⁡ ( C ) → S with Spec ⁡ ( σ ) . Given ( V B , V ét , V dR ) ∈ R ⁢ ( S ) , we write σ V ét , s : V B , s ⊗ A f ⁢ → ∼ ⁢ V B , σ ⁢ ( s ) ⊗ A f for the natural isomorphism induced by V ét , viewed as a descent of V B ⊗ A f from S C to 𝑆; similarly, we write σ V dR , s : V B , s ⊗ C ⁢ → ∼ ⁢ V B , σ ⁢ ( s ) ⊗ C for the natural 𝜎-linear isomorphism induced by the descent V dR of (the algebraization of) V B ⊗ O S C an .

To clarify the meaning of σ V ét , s , we remark that 𝑠 and σ ⁢ ( s ) are usually different closed points on S C , and they are equal if and only if 𝜎 fixes the residue field k ⁢ ( s 0 ) , where s 0 ∈ S is the image of 𝑠. The collection of σ V ét , s as 𝜎 runs through Aut ⁢ ( k ⁢ ( s ) / k ⁢ ( s 0 ) ) = Aut ⁢ ( C / k ⁢ ( s 0 ) ) is nothing but the Galois action on the stalk V B , s ⊗ A f = V ét , s .

Using the notation of Section 2.1.3 and Notation 3.1.4, we define the following.

Definition 3.1.5

We say that a system of realizations 𝖵 is weakly abelian-motivic (weakly AM) if, for every s ∈ S ⁢ ( C ) , there exists M ∈ Mot Ab ⁢ ( C ) such that

ω Hdg ⁢ ( M ) ≅ ( V B , s , V dR , s ) ;

moreover, for any such isomorphism γ : ω Hdg ⁢ ( M ) ⁢ → ∼ ⁢ ( V B , s , V dR , s ) and σ ∈ Aut ⁢ ( C / k ) , there exists an isomorphism γ σ : ω Hdg ⁢ ( M σ ) ⁢ → ∼ ⁢ ( V B , σ ⁢ ( s ) , V dR , σ ⁢ ( s ) ) such that the Betti components γ B , γ B σ of γ , γ σ fit into commutative diagrams

We note that γ σ is uniquely determined by 𝛾 provided that it exists. Denote the full subcategory of R ⁢ ( S ) given by these objects by R am ∗ ⁢ ( S ) . It is easy to check that if T → S is a morphism between smooth 𝑘-varieties, the natural pullback functor R ⁢ ( S ) → R ⁢ ( T ) sends R am ∗ ⁢ ( S ) to R am ∗ ⁢ ( T ) .

Lemma 3.1.6

Lemma 3.1.6 ([63, (3.4.1)])

Suppose that 𝑆 is a smooth variety over k ⊆ C and take V , W ∈ R am ∗ ⁢ ( S ) . Let φ C be a morphism V | S C → W | S C . Then φ C descends to a morphism V → W if and only if either the étale or the de Rham component of φ C descends to 𝑆.

When applied to the case V = 1 S , the above lemma says that, for W ∈ R am ∗ ⁢ ( S ) , if the étale realization of a global section of W B | S C which is everywhere of Hodge type ( 0 , 0 ) descends to 𝑆, then so does the de Rham realization, and vice versa. This is a global version of the following statement. Suppose that M ∈ Mot AH ⁢ ( k ) and that v ∈ ω B ⁢ ( M C ) is absolute Hodge. Then v ⊗ A f ∈ ω ét ⁢ ( M C ) descends to 𝑘 (i.e., is Aut ⁢ ( C / k ) -invariant) if and only if v ⊗ C ∈ ω dR ⁢ ( M C ) descends to ω dR ⁢ ( M ) . Note that if M ∈ Mot Ab ⁢ ( k ) , then any Hodge cycle v ∈ ω B ⁢ ( M C ) is automatically absolute Hodge.

For future reference, we give a handy lemma on Definition 3.1.2 (b) and (c).

Lemma 3.1.7

Suppose that, in Definition 3.1.2 (a), the representation V ∈ Rep ⁢ ( G ) is faithful, 𝑆 is geometrically connected as a 𝑘-variety, and for some b ∈ S ⁢ ( C ) , there is an isomorphism η b : V ⁢ → ∼ ⁢ V B , b such that [ η ] b = K ⋅ ( η b ⊗ A f ) ; moreover, for some Ω as in Definition 3.1.2 (c), the Hodge structure on V B , b is defined by an element of Ω via η b . Then, assuming either k = C or V ∈ R am ∗ ⁢ ( S ) , [ η ] is 𝖵-rational and of type Ω.

Proof

Note that, by assumption, S C is connected, and the fact that η b ⊗ A f represents a 𝖪-level structure at 𝑏 implies that its 𝖪-orbit is π 1 ét ⁢ ( S , b ) -stable. In particular, for each v ∈ ( V ⊗ ) G , π 1 ⁢ ( S C , b ) fixes η b ⁢ ( v ) , so there exists a v B ∈ H 0 ⁢ ( V B ⊗ ) such that v B , b = η b ⁢ ( v ) . Likewise, there exists v ét ∈ H 0 ⁢ ( V ét ⊗ ) such that v ét , b = η b ⁢ ( v ) ⊗ 1 , and v ét is precisely the η ⁢ ( v ) ét in Definition 3.1.2 (b).

Note that v B is necessarily of Hodge type ( 0 , 0 ) at 𝑏, because the Mumford–Tate group of the Hodge structure ( V B , b , V dR , b ) is contained in 𝐺 via η b . This implies that v B is of Hodge type ( 0 , 0 ) everywhere by the theorem of the fixed part. Let v dR , C be the global section in ( V B ⊗ O S C an ) ⊗ induced by v B . Then v dR , C algebraizes to a global section of ( V dR | S C ) ⊗ (cf. [10, II, Theorem 5.9]).

If k = C (so that S = S C ), then we have already shown that [ η ] is 𝖵-rational; moreover, one deduces from the connectedness of S C and [13, (1.1.12)] that [ η ] remains of type Ω on S C . If k ⊆ C is a general subfield, it remains to show that v dR , C descends to 𝑆 under the additional assumption that V ∈ R am ∗ ⁢ ( S ) . However, as the étale realization of v B descends to a global section of V ét ⊗ (i.e., v ét ), this follows from Lemma 3.1.6. ∎

3.2 Shimura varieties

Let ( G , Ω ) be a Shimura datum which satisfies the axioms in [45, II, (2.1)]. Let E ⁢ ( G , Ω ) be the reflex field, 𝑍 the center of 𝐺, and Z s the maximal anisotropic subtorus of 𝑍 that is split over ℝ. In this paper, we always assume that

(3.1) the weight is defined over ⁢ Q ⁢ and ⁢ Z s ⁢ is trivial .

Note that, in particular, the latter condition ensures that Z ⁢ ( Q ) is discrete in Z ⁢ ( A f ) (see [47, Remark 5.27]). We will often drop the Hermitian symmetric domain Ω from the notation of Shimura varieties when no confusion would arise.

For any compact open subgroup K ⊆ G ⁢ ( A f ) , let Sh K ⁢ ( G ) C denote the resulting Shimura variety with a complex uniformization

G ⁢ ( Q ) \ Ω × G ⁢ ( A f ) / K

and let Sh K ⁢ ( G ) denote the canonical model over E ⁢ ( G , Ω ) . Let Sh ⁢ ( G ) denote the inverse limit lim ← K ⁡ Sh K ⁢ ( G ) as 𝖪 runs through all compact open subgroups. Under our assumptions, Sh ⁢ ( G ) ⁢ ( C ) is described by G ⁢ ( Q ) \ Ω × G ⁢ ( A f ) (see [47, (5.28)]). Note that Sh K ⁢ ( G ) = Sh ⁢ ( G ) / K .

3.2.1

Let G → GL ⁢ ( V ) be a representation. We can attach to Sh K ⁢ ( G ) C for any neat compact open subgroup K ⊆ G ⁢ ( A f ) an automorphic VHS ( V B , V dR , C ) (cf. [45, Chapter II, 3.3], [57, §2.2]). In particular, V B is defined to be the contraction product

V B : = V × G ⁢ ( Q ) [ Ω × G ( A f ) / K ] .

The filtration on V dR , C is obtained by descending the filtration on the tautological VHS on V × Ω . Analogously, the automorphic étale local system V ét on the proétale site of Sh K ⁢ ( G ) is defined as the contraction product V ét : = V A f × K Sh ( G ) which comes with a comparison isomorphism V B ⊗ A f ≅ V ét | Sh K ⁢ ( G ) C . Moreover, by construction, V ét over Sh K ⁢ ( G ) comes with a tautological 𝖪-level structure [ η V ] , i.e., a global section of K \ Isom ¯ ⁢ ( V ⊗ A f , V ét ) .

Define [ η V an ] to be the restriction of [ η V ] to Sh K ⁢ ( G ) C . Using ( V B , V dR , C ) , we can already give a moduli interpretation of Sh K ⁢ ( G ) C . Below is a reformulation of [46, Proposition 3.10] in our terminology (cf. [63, (4.1.2)])^[5].

Theorem 3.2.2

Assume that 𝑉 is faithful. For every smooth ℂ-variety 𝑇, let M V ⁢ ( T ) be the groupoid of tuples of the form ( W , [ ξ an ] ) , where W = ( W B , W dR ) is a VHS over 𝑇 and [ ξ an ] is a 𝖪-level structure on W B ⊗ A f which is 𝖶-rational and is of type Ω. Then

( V C : = ( V B , V dR , C ) , [ η V an ] )

is an object of M V ⁢ ( Sh K ⁢ ( G ) C ) , and for every object ( W , [ ξ an ] ) ∈ M V ⁢ ( T ) , there exists a unique morphism ρ : T → Sh K ⁢ ( G ) C such that ρ ∗ ⁢ ( V C , [ η V an ] ) ≅ ( W , [ ξ an ] ) .

We remark that, as 𝖪 is neat, assumption (3.1) ensures that the objects in M V ⁢ ( T ) above have no nontrivial automorphisms (cf. [46, Remark 3.11]).

To describe the moduli problem for the canonical model, we restrict to the following subclass of Shimura data, which contains all cases we will consider in the following chapters.

Assumption 3.2.3

There exists a morphism of Shimura data ( G ̃ , Ω ̃ ) → ( G , Ω ) such that

( G ̃ , Ω ̃ ) is of Hodge type and also satisfies assumption (3.1);
G ̃ → G is surjective and the kernel lies in the center of G ̃ ;
the embedding of reflex fields E ⁢ ( G , Ω ) ⊆ E ⁢ ( G ̃ , Ω ̃ ) is an equality.

Below, we assume that ( G , Ω ) is a Shimura datum which satisfies the above assumptions.

3.2.4

For any representation G → GL ⁢ ( V ) , and V B , V dR , C , V ét set up in Section 3.2.1, there exists a unique descent V dR of V dR , C to Sh K ⁢ ( G ) such that ( V B , V dR , V ét ) is a weakly AM system of realizations (see [63, (4.2.2)]). We call V : = ( V B , V dR , V ét ) ∈ R am ∗ ( Sh K ( G ) ) the automorphic system (of realizations) on Sh K ⁢ ( G ) attached to V ∈ Rep ⁢ ( G ) . The tautological 𝖪-level structure [ η V ] on 𝖵 is 𝖵-rational, and is of type Ω when 𝑉 is faithful.

Theorem 3.2.5

Theorem 3.2.5 ([63, (4.3.1)])

Assume that 𝑉 is faithful. Let E ′ ⊆ C be a subfield which contains E = E ⁢ ( G , Ω ) and let 𝑇 be a smooth E ′ -variety. Let M V , E ′ ⁢ ( T ) be the groupoid of pairs ( W , [ ξ ] ) , where W = ( W B , W dR , W ét ) ∈ R am ∗ ⁢ ( T ) , and [ ξ ] is a 𝖪-level structure on W ét such that [ ξ ] is 𝖶-rational and ( W , [ ξ ] ) is of type Ω. Then ( V = ( V B , V dR , V ét ) , [ η V ] ) is an object of M V , E ⁢ ( Sh K ⁢ ( G ) ) , and for each ( W , [ ξ ] ) ∈ M V , E ′ ⁢ ( T ) , there exists a unique ρ : T → Sh K ⁢ ( G ) E ′ such that ( W , [ ξ ] ) ≅ ρ ∗ ⁢ ( V , [ η V ] ) .

Note that any object in M V , E ′ ⁢ ( T ) above has no nontrivial automorphisms because this is already true when E ′ = C . This is a key fact that we shall use repeatedly throughout the paper. The above is proved by first constructing a morphism ρ C : T C → Sh K ⁢ ( G ) C such that ( W , [ ξ ] ) | T C ≅ ρ C ∗ ⁢ ( V , [ η V ] ) and then showing that the action of ρ C on the ℂ-points is Aut ⁢ ( C / E ′ ) -equivariant. This is clearly inspired by the proof of [40, Corollary 5.4]. However, unlike [40, Proposition 5.6 (1)], we show that ( W , [ ξ ] ) | T C ≅ ρ C ∗ ⁢ ( V , [ η V ] ) descends over 𝑇 using a rigidity lemma about weakly AM systems of realizations (see [63, (3.4.4), (3.4.5), and (4.3.2)]).

3.3 Orthogonal Shimura varieties over totally real fields

3.3.1

Let 𝐸 be a totally real number field and let 𝒱 be a quadratic form over 𝐸 which has signature ( 2 , dim E V − 2 ) at a unique real place 𝜏 and is negative definitive at every other real place. We set G : = Res E / Q SO ( V ) and

Ω V : = { v ∈ P ( V ⊗ τ C ) ∣ ⟨ v , v ⟩ = 0 , ⟨ v , v ̄ ⟩ > 0 } .

Let 𝑉 be the transfer V ( Q ) (recall Notation 2.1.5). Set G : = SO ( V ) and define Ω : = Ω V as above (applied to the E = Q case). Note that 𝒢 is the identity component of the centralizer of the 𝐸-action in 𝐺. We view Ω as a G ⁢ ( R ) -conjugacy class of morphisms S → G R such that an element v ∈ Ω corresponds to the morphism ℎ which gives 𝑉 a Hodge structure of K3-type with v = V C ( 1 , − 1 ) (cf. [41, §3.1]). Likewise, we view Ω V as a G ⁢ ( R ) -conjugacy class of morphisms S → G R , which are those whose composition with G R → G R defines a Hodge structure on 𝑉 which is preserved by the 𝐸-action on V = V ( Q ) . It is well known that ( G , Ω V ) is a Shimura datum with reflex field 𝐸, viewed as a subfield of ℂ under 𝜏.

Let G ̃ denote CSpin ⁢ ( V ) . Then ( G ̃ , Ω ) is a Shimura datum of Hodge type with reflex field ℚ which admits a natural morphism to ( G , Ω ) .

Lemma 3.3.2

( G , Ω V ) satisfies Assumption 3.2.3.

Proof

Consider G ̃ ′ : = Res E / Q CSpin ( V ) . It is well known that ( G ̃ ′ , Ω V ) is a Shimura datum with reflex field 𝐸. Unfortunately, it does not satisfy condition (3.1) unless E = Q as the center Z ′ of G ̃ ′ has identity component Res E / Q ⁡ G m , E . Thus we modify this approach by dividing out the maximal anisotropic torus of Res E / Q ⁡ G m , E ; explicitly, we consider ( Nm is the norm map)

G ̃ : = G ̃ ′ / ker ( Nm : Res E / Q G m , E → G m ) ) .

Note that ker ⁡ ( G ̃ ′ → G ) = Res E / Q ⁡ G m , E , so we obtain a morphism ( G ̃ , Ω V ) → ( G , Ω V ) of Shimura data. It remains to check Assumption 3.2.3 (i) and (iii). First, note that G ̃ can be canonically identified with the fiber product G × G G ̃ (see [49, §§4.2, 4.3]), so that there is a diagram with exact rows

Now Z s ⁢ ( G ̃ ) = 1 is clear. As ( G ̃ , Ω ) is of Hodge type and ( G ̃ , Ω V ) embeds into ( G ̃ , Ω ) , ( G ̃ , Ω V ) is also of Hodge type. As the reflex fields of ( G , Ω V ) and ( G ̃ ′ , Ω V ) are both well known to be 𝐸, the same must be true for ( G ̃ , Ω V ) . ∎

3.3.3

We do some preparations to future reference. Below, for any ℚ-linear Tannakian category 𝒞, we write N : Mod E ⁢ ( C ) → C for either one of the following two functors:

M ↦ Nm E / Q ⁢ ( M ) or M ↦ Nm E / Q ⁢ ( M ) ⊗ Q det ( M ( Q ) )

(see notation in Section 2.1.4). Any conclusion applies to both functors.

Recall Notation 2.1.5, and write Z ⁢ ( V ) and H ⁢ ( V ) simply as 𝒵 and ℋ. Then N : = N ( V ) is a faithful representation of ℋ, as the composite G → Res E / Q ⁡ GL ⁢ ( V ) → GL ⁢ ( N ) has kernel exactly 𝒵. Let K ⊆ G ⁢ ( A f ) and C ⊆ H ⁢ ( A f ) be compact open subgroups such that 𝒞 contains the image of 𝒦. For each prime ℓ, let K ℓ be the image of 𝒦 under the projection G ⁢ ( A f ) → G ⁢ ( Q ℓ ) .

Let Ω H be the H ⁢ ( R ) -conjugacy class of morphisms S → H R which contains the image of Ω V . Then it follows that ( H , Ω H ) is a Shimura datum which admits a natural morphism ( G , Ω V ) → ( H , Ω H ) . Since 𝒵 lies in the center of 𝒢 and is discrete, by [12, Proposition 3.8], ( H , Ω H ) has the same reflex field as ( G , Ω V ) . Moreover, one easily checks that ( H , Ω H ) satisfies Assumption 3.2.3 using that ( G , Ω V ) does.

Remark 3.3.4

Note that if dim E V is odd, then 𝒵 is trivial; in this case, the reader should read the content below with ( G , Ω V , K ) = ( H , Ω H , C ) in mind. The case when dim E V is even will only be used for Section 4.3.

3.3.5

Let k ⊆ C be a subfield and 𝑇 a smooth 𝑘-variety. Note that the identification V = V ( Q ) gives 𝑉 an 𝐸-action, which commutes with 𝒢. Suppose that W ∈ R ⁢ ( T ) is a system equipped with a ( G , V , K ) -level structure [ μ ] . Then [ μ ] transports the 𝐸-action on V ⊗ A f to one on W ét , through which we may view [ μ ] as a global section of K \ Isom ¯ E ⁢ ( V ⊗ Q A f , W ét ) . Moreover, there is a natural map

K \ Isom ¯ E ⁢ ( V ⊗ Q A f , W ét ) → C \ Isom ¯ ⁢ ( N ⁢ ( V ) ⊗ A f , N ⁢ ( W ét ) )

through which [ μ ] defines an ( H , N , C ) -level structure on N ⁢ ( W ét ) , which we denote by N ⁢ ( [ μ ] ) .

Lemma 3.3.6

Let 𝑇, 𝖶, [ μ ] be as above. Let W ′ ∈ R ⁢ ( T ) be another system with 𝒦-level structure [ μ ′ ] . Suppose that there is an isomorphism γ ét : ( W ét , [ μ ] ) | T C ≅ ( W ét ′ , [ μ ′ ] ) | T C such that N ⁢ ( γ ét ) descends to an isomorphism ( N ⁢ ( W ét ) , N ⁢ ( [ μ ] ) ) ≅ ( N ⁢ ( W ét ′ ) , N ⁢ ( [ μ ′ ] ) ) over 𝑇. If ℓ is a prime such that K ℓ ∩ Z ⁢ ( Q ℓ ) = 1 , then the ℓ-adic component γ ℓ of γ ét descends to an isomorphism W ℓ ≅ W ℓ ′ over 𝑇.

Proof

We may assume 𝑇 is connected. Choose a base point b ∈ T ⁢ ( C ) . Set γ : = γ ℓ , b and let g ∈ π 1 ét ⁢ ( T , b ) be any element. Our goal is to show that δ : = g − 1 γ − 1 g γ = 1 . Note that, as N ⁢ ( γ ét ) descends to 𝑇, we already know that N ⁢ ( δ ) = 1 . Let μ : V ⊗ Q A f ⁢ → ∼ ⁢ W ét , b be a representative of [ μ ] b . Then μ − 1 ⁢ g ⁢ μ ∈ K ℓ , because the 𝒦-orbit [ μ ] b is π 1 ét ⁢ ( S , b ) -stable. Set μ ′ : = γ μ . Then μ ′ represents [ μ ′ ] b , so that ( μ ′ ) − 1 ⁢ g ⁢ μ ′ ∈ K ℓ . Now we have

μ − 1 ⁢ δ ⁢ μ = [ μ − 1 ⁢ g − 1 ⁢ μ ] ⁢ [ ( μ ′ ) − 1 ⁢ g ⁢ μ ′ ] ∈ K ℓ .

Note that N ⁢ ( μ − 1 ⁢ δ ⁢ μ ) = 1 ∈ GL ⁢ ( N ⊗ Q ℓ ) . However, as the kernel of the K ℓ -action on N ⊗ Q ℓ lies in Z ⁢ ( Q ℓ ) , we must have μ − 1 ⁢ δ ⁢ μ = 1 , i.e., δ = 1 . ∎

3.3.7

We will often consider the following diagram of Shimura data:

Let K ⊆ G ⁢ ( A f ) , K ⊆ G ⁢ ( A f ) , and C ⊆ H ⁢ ( A f ) be neat compact open subgroups such that K ⊆ K and 𝒞 contains the image of 𝒦. Then we have Shimura morphisms

i : Sh K ⁢ ( G ) → Sh K ⁢ ( G ) E and π : Sh K ⁢ ( G ) → Sh C ⁢ ( H )

defined over 𝐸. Let 𝖵 (resp. V ̃ ) be the automorphic system on Sh K ⁢ ( G ) (resp. Sh K ⁢ ( G ) ) attached to the standard representation 𝑉 of 𝐺 (resp. 𝒢) and let [ η V ] (resp. [ η V ] ) be the tautological 𝖪-level structure (resp. 𝒦-level structure), as defined in Section 3.2.4. Then there is a natural identification V ̃ = i ∗ ⁢ ( V ) and [ η V ] refines i ∗ ⁢ ( [ η V ] ) in the sense of Definition 3.1.2 (d).

Note that [ η V ] endows V ̃ with a canonical 𝐸-action. A priori, it only defines an 𝐸-action on V ̃ ét , but since [ η V ] is V ̃ -rational, its restriction to Sh K ⁢ ( G ) C comes from an 𝐸-action on V ̃ B . Then, by Lemma 3.1.6, one deduces that the resulting 𝐸-action on V ̃ dR | Sh K ⁢ ( G ) C via the Riemann–Hilbert correspondence descends to Sh K ⁢ ( G ) . Therefore, it makes sense to form N ⁢ ( V ̃ ) . Since V ̃ is weakly AM in the sense of Definition 3.1.5, so is N ⁢ ( V ̃ ) . This is simply because we may apply the functor N ⁢ ( − ) to Mod E ⁢ ( Mot Ab ⁢ ( C ) ) and it commutes with the cohomological realizations.

Now let 𝖭 be the automorphic system on Sh C ⁢ ( H ) attached to N ∈ Rep ⁢ ( H ) and let [ η N ] be its tautological 𝒞-level structure. We claim that there is an (necessarily unique) isomorphism

(3.2) π ∗ ⁢ ( N , [ η N ] ) ≅ ( N ⁢ ( V ̃ ) , N ⁢ ( [ η V ] ) ) .

Indeed, to verify this isomorphism, one first checks its implications on the automorphic VHS and étale local systems, which follow from the explicit descriptions in Section 3.2.1, then applies Lemma 3.1.6.

3.4 Integral model

Let 𝑉 be a quadratic form over ℚ and suppose that there is a self-dual Z ( p ) -lattice L ( p ) ⊆ V for a prime p > 2 . Then G : = SO ( V ) extends to the reductive Z ( p ) -group SO ⁢ ( L ( p ) ) , which we still write as 𝐺 by abuse of notation. Let K ⊆ G ⁢ ( A f ) be a neat compact open subgroup of the form K p ⁢ K p with K p = G ⁢ ( Z p ) and K p ⊆ G ⁢ ( A f p ) . Then, by [33, 41], Sh K ⁢ ( G ) admits a canonical integral model S K ⁢ ( G ) over Z ( p ) .

The Shimura variety S K ⁢ ( G ) is typically studied via the corresponding spinor Shimura variety, which is of Hodge type. Let G ̃ : = CSpin ( L ( p ) ) . Set K p to be G ̃ ⁢ ( Z p ) , K p ⊆ G ̃ ⁢ ( A f p ) to be a small enough compact open subgroup whose image in G ⁢ ( A f p ) is contained in K p , and 𝕂 to be the product K p ⁢ K p . The reflex field of ( G ̃ , Ω ) is ℚ, and by [33], there is an canonical integral model S K ⁢ ( G ̃ ) over Z ( p ) . There are a suitable symplectic space ( H , ψ ) and a Siegel half-space H ± such that there is an embedding of Shimura data ( G ̃ , Ω ) ↪ ( GSp ⁢ ( ψ ) , H ± ) which eventually equips S K ⁢ ( G ̃ ) with a universal abelian scheme 𝒜.^[6] Let a : A → S K ⁢ ( G ̃ ) be the structural morphism. Define the sheaves

H B := R 1 ⁢ a C ⁣ ∗ ⁢ Z ( p ) , H ℓ := R 1 ⁢ a ∗ ⁢ Q ¯ ℓ ⁢ ( ℓ ≠ p ) , H p := R 1 ⁢ a Q ⁣ ∗ ⁢ Z ¯ p , H dR := R 1 ⁢ a ∗ ⁢ Ω A / S K ⁢ ( G ̃ ) ∙ , and H cris := R 1 ⁢ a ̄ cris ⁣ ∗ ⁢ O A F p / Z p ⁢ ( a ̄ := a ⊗ F p ) .

The abelian scheme 𝒜 is equipped with a “CSpin-structure”: a Z / 2 ⁢ Z -grading, Cl ⁢ ( L ) -action, and an idempotent projector π ? : End ⁡ ( H ? ) → End ⁡ ( H ? ) for ? = B , ℓ , p , dR , cris on (various applicable fibers of) S K ⁢ ( G ̃ ) . We use L ? to denote the images of π ? , and recall the definition of special endomorphisms [41, Definition 5.2; see also Lemma 5.4, Corollary 5.22].

Definition 3.4.1

For any S K ⁢ ( G ̃ ) -scheme 𝑇, f ∈ End ⁡ ( A T ) is called a special endomorphism if, for some (and hence all) ℓ ∈ O T × , the ℓ-adic realization of 𝑓 lies in

L ℓ | T ⊆ End ⁡ ( H ℓ | T ) ;

if O T = k for a perfect field 𝑘 in characteristic 𝑝, then equivalently 𝑓 is called a special endomorphism if the crystalline realization of 𝑓 lies in L cris , T . We write the submodule of End ⁡ ( A T ) consisting of special endomorphisms as L ⁢ ( A T ) .

3.4.2

It is explained in [41, §5.24] that the sheaves L ? on (applicable fibers of) S K ⁢ ( G ̃ ) in fact descend to the corresponding fibers of S K ⁢ ( G ) . We denote the descent of these sheaves by the same letters. It is not hard to see that H B ⁢ [ 1 / p ] together with the restrictions of

∏ ℓ ≠ p H ℓ × H p ⁢ [ 1 / p ]

and H dR to Sh K ⁢ ( G ̃ ) is nothing but the automorphic system attached to H ∈ Rep ⁢ ( G ̃ ) , in our terminology of Section 3.2.4. Similarly, L B ⁢ [ 1 / p ] together with the restrictions of

∏ ℓ ≠ p L ℓ × L p ⁢ [ 1 / p ]

and L dR to Sh K ⁢ ( G ̃ ) (or Sh K ⁢ ( G ) ) is precisely the automorphic system attached to V ∈ Rep ⁢ ( G ̃ ) (or Rep ⁢ ( G ) ). Readers who wish to check this can look at how the automorphic systems are constructed in [63, §4.2], which is essentially a generalization of [41, §5.24]. The construction of the L ? -sheaves is also summarized in more detail in [62, (3.1.3)]. In particular, there are natural identifications of V B with L B ⁢ [ 1 / p ] , and V ét (resp. V dR ) with the restriction of ∏ ℓ ≠ p L ℓ × L p ⁢ [ 1 / p ] (resp. L dR ) to Sh K ⁢ ( G ) , where ( V B , V dR , V ét ) was the notation we used in Section 3.3.7.

3.4.3

Let Sh K p ⁢ ( G ) be the limit lim ← K p ⁡ Sh K p ⁢ K ′ ⁣ p ⁢ ( G ) as K ′ ⁣ p runs through the compact open subgroups of G ⁢ ( A f p ) and define S K p ⁢ ( G ) similarly. The canonical extension property of S K p ⁢ ( G ) is that every morphism S Q → S K p ⁢ ( G ) extends to 𝑆 for any regular, formally smooth Z ( p ) -scheme 𝑆. We give an extension property for finite level, which is certainly well known to experts.

Theorem 3.4.4

Let 𝑆 be a smooth Z ( p ) -scheme which admits a morphism

ρ : S Q → Sh K ⁢ ( G ) .

If ρ ∗ ⁢ V ℓ extends to a local system W ℓ over 𝑆 for every prime ℓ ≠ p , then 𝜌 extends to a morphism S → S K ⁢ ( G ) , through which W ℓ is identified with the pullback of L ℓ .

Proof

To simplify notation, let V ét ( p ) : = ∏ ℓ ≠ p V ℓ and W ét ( p ) : = ∏ ℓ ≠ p W ℓ . Note that W ét ( p ) | S Q = ρ ∗ ⁢ ( V ét ( p ) ) . Now the prime-to-𝑝 part of the tautological level structure [ η V ] gives us a section

[ η V ( p ) ] : Sh K ⁢ ( G ) → K p \ Isom ¯ ⁢ ( V ⊗ A f p , V ét ( p ) ) ,

where the target is viewed as a pro-étale cover of Sh K ⁢ ( G ) . By [66, Tag 0BQM], ρ ∗ ⁢ [ η V ( p ) ] extends to a morphism

S → K p \ Isom ¯ ⁢ ( V ⊗ A f p , W ét ( p ) ) ,

where the target is a pro-étale cover of 𝑆. We define S ̃ as the pullback

Note that, since S ̃ → S is a K p -torsor for the proétale topology, S ̃ is representable by a scheme. As Sh K p ⁢ ( G ) → Sh K ⁢ ( G ) can be defined by an analogous construction, 𝜌 lifts to a K p -equivariant morphism ρ ̃ : S ̃ Q → Sh K p ⁢ ( G ) . The canonical extension property allows us to extend it to a (necessarily K p -equivariant) morphism S ̃ → S K p ⁢ ( G ) . Now the K p -action defines an étale descend datum which yields the desired morphism S → S K ⁢ ( G ) . ∎

4 Period morphisms

4.1 The basic set-up

We first state a few basic definitions and results which will be needed to construct the period morphism.

Definition 4.1.1

Let 𝑇 be a connected Noetherian normal scheme with geometric point 𝑡, let ℓ be a prime with ℓ ∈ O T × , and let W ℓ be an étale Q ℓ -local system over 𝑇. We denote by Mon ⁢ ( W ℓ , t ) the Zariski closure of the image of π 1 ét ⁢ ( T , t ) in GL ⁢ ( W ℓ , t ) , and by Mon ∘ ⁢ ( W ℓ , t ) the identity component of Mon ⁢ ( W ℓ , t ) .

If W ℓ ′ is another Q ℓ -local system, then we say that W ℓ and W ℓ ′ are étale locally isomorphic if they are isomorphic over some finite connected étale cover T ′ of 𝑇, or equivalently, there is an isomorphism W ℓ , t ⁢ → ∼ ⁢ W ℓ , t ′ which is equivariant under an open subgroup of π 1 ét ⁢ ( T , t ) .

Lemma 4.1.2

Let 𝑇 be a Noetherian integral normal scheme with generic point 𝜂. Let f : Y → T be a smooth proper morphism.

The natural map Pic ⁡ ( Y ) → Pic ⁡ ( Y η ) is surjective with kernel im ⁡ ( Pic ⁡ ( T ) → Pic ⁡ ( Y ) ) .
If, for some geometric point 𝑏 over 𝜂 and prime ℓ ∈ O T × , Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q ℓ , b ) is connected, then the natural map NS ⁢ ( Y η ) Q → NS ⁢ ( Y b ) Q is an isomorphism.

Proof

(a) We may always extend a line bundle on Y η to Y U for some open dense subscheme U ⊆ T , and then to a line bundle on 𝒴 (use, e.g., [25, Proposition II.6.5]). But any two extensions to 𝒴 differ by an element of Pic ⁡ ( T ) by [22, Err_IV, Corollary 21.4.13].

(b) Let η ̄ be the geometric point over 𝜂 obtained by taking the separable closure of k ⁢ ( η ) in k ⁢ ( b ) . Then every class of NS ⁢ ( Y b ) Q descends to η ̄ . As the natural morphism

Gal ⁢ ( η ̄ / η ) = π 1 ét ⁢ ( η , b ) → π 1 ét ⁢ ( T , b )

is surjective, and Gal ⁢ ( η ̄ / η ) acts on NS ⁢ ( Y η ̄ ) Q through a finite quotient, the connectedness assumption on Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q ℓ , b ) implies that Gal ⁢ ( η ̄ / η ) in fact acts trivially. This implies that every class in NS ⁢ ( Y η ̄ ) Q descends to 𝜂. ∎

Now we state the set-up we will work with for the entire section.

Set-up 4.1.3

Let F ⊆ C be a subfield finitely generated over ℚ and let 𝑆 be a connected smooth 𝐹-variety with generic point 𝜂. Let f : X → S be a ♡ -family with a relatively ample line bundle 𝝃. We fix a subspace Λ ⊆ NS ⁢ ( X η ) Q containing the class of ξ η . Now choose an 𝐹-linear embedding k ⁢ ( η ) ↪ C and let b ∈ S ⁢ ( C ) be the resulting point, which we also view as a closed point on S C . Let S ∘ ⊂ S C denote the connected component containing 𝑏. We assume that, for some (and hence every; see Lemma 4.1.4 below) prime ℓ, Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q ℓ , b ) is connected.

Define a pairing on H dR 2 ( X / S ) : = R 2 f ∗ Ω X / S ∙ by ( x , y ) ↦ x ∪ y ∪ c 1 ⁢ ( ξ ) d − 2 for local sections x , y and d = dim X / S . By Lemma 4.1.2, every line bundle on X η extends to a relative line bundle on X / S . By taking Chern classes, we obtain a well defined embedding Λ ¯ ↪ H dR 2 ⁢ ( X / S ) , where Λ ¯ is the constant sheaf with fiber Λ. Define P dR to be the orthogonal complement of Λ ¯ in H dR 2 ⁢ ( X / S ) ⁢ ( 1 ) . We define the primitive Betti cohomology P B over S C and étale cohomology P ét over 𝑆 analogously. Then P : = ( P B , P dR , P ét ) ∈ R ( S ) is a system of realizations over 𝑆 in the sense of Definition 3.1.1. Note that we applied a Tate twist so that the VHS P | S C has weight 0. Since we assumed Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q ℓ , b ) is connected for some ℓ, NS ⁢ ( X η ) Q = NS ⁢ ( X b ) Q by Lemma 4.1.2. This implies that 𝖯 orthogonally decomposes into ( Λ ⟂ ⊗ 1 S ) ⊕ P 0 , where Λ ⟂ is the orthogonal complement of Λ in NS ⁢ ( X η ) Q and P 0 = ( P 0 , B , P 0 , dR , P 0 , ét ) is another object in R ⁢ ( S ) . Moreover, there are no nonzero ( 0 , 0 ) -classes in P 0 , B , b . Note that P 0 is nothing but 𝖯 when Λ = NS ⁢ ( X η ) Q .

Lemma 4.1.4

In the above set-up, Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q ℓ , b ) is connected for every prime ℓ, and 𝑏 is a Hodge-generic point for the VHS P | S ∘ .

Proof

The first statement follows from the assumption that 𝐹 is finitely generated over ℚ and [38, Proposition 6.14], which implies that the étale group scheme of connected components of Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q ℓ , b ) is independent of ℓ. The second statement follows from the main theorem of [49].^[7] Since the statement only concerns S ∘ , we may replace 𝑆 by S ∘ and 𝐹 by its field of definition and thus assume that 𝑆 is geometrically connected. Then we may make use of the notion of Galois-generic points [48, Definition 4.2.1]. As the Mumford–Tate conjecture is known for H 2 of the fibers of X / S and 𝑏 lies above 𝜂, the fact that 𝜂 is Galois-generic implies that 𝑏 is Hodge-generic. ∎

4.1.5

Let 𝑉 be a quadratic form over ℚ which is isomorphic to P B , b and fix an isometry μ b : V ⁢ → ∼ ⁢ P B , b . Let G : = SO ( V ) . Let V 0 : = μ b − 1 ( P 0 , B , b ) . Note that, via μ b , the monodromy representation of π 1 ét ⁢ ( S , b ) takes values in G ⁢ ( A f ) . We say that 𝖪 is admissible if the image of π 1 ét ⁢ ( S , b ) in GL ⁢ ( P B , b ⊗ A f ) lies in 𝖪 via μ b , or equivalently, μ b extends to a ( G , V , K ) -level structure [ μ ] on P ét with [ μ ] b = K ⋅ ( μ b ⊗ A f ) . Define

Ω : = { v ∈ P ( V ⊗ C ) ∣ ⟨ v , v ⟩ = 0 , ⟨ v , v ̄ ⟩ > 0 }

as in Section 3.3.3. Then ( G , Ω ) is a Shimura datum of abelian type with reflex field ℚ. Again, let V = ( V B , V dR , V ét ) be the automorphic system on Sh K ⁢ ( G ) attached to V ∈ Rep ⁢ ( G ) , and let [ η V ] be the tautological 𝖪-level structure on V ét (see Section 3.2.4).

4.1.6

Suppose ( X / S , ξ ) | S ∘ belongs to case (R+) or (R2^′) described in Section 2.2.4, i.e., the endomorphism field 𝐸 of the Hodge structure on P 0 , B , b is totally real. By Theorem 2.2.2, the 𝐸-action on P 0 , B , b is self-adjoint. Let 𝐸 act on V 0 through μ b . Recall Notation 2.1.5. Let 𝒱 be the 𝐸-bilinear lift of V 0 , i.e., V 0 = V ( Q ) and set G : = Res E / Q SO ( V ) , H : = G / Z E 1 . In addition, set V ♯ : = V ⊕ E and G ♯ : = Res E / Q SO ( V ♯ ) . Embed SO ⁢ ( V ) into SO ⁢ ( V ♯ ) by acting trivially on ℰ. This induces an embedding G ↪ G ♯ .

We say that a neat compact open subgroup K ⊂ G ⁢ ( A f ) satisfies condition (♯) depending on the particular case ( K : = K ∩ G ( A f ) below):

always;
if the image of 𝒦 in G ♯ ⁢ ( A f ) lies in some neat compact open subgroup;
if the image of 𝒦 in H ⁢ ( A f ) lies in some neat compact open subgroup.

In case (R2^′), we say that, for a prime ℓ 0 , K ℓ 0 is sufficiently small if K ℓ 0 ∩ Z E 1 ⁢ ( Q ℓ 0 ) = 1 . Here K ℓ 0 is the image of 𝖪 under the projection G ⁢ ( A f ) → G ⁢ ( Q ℓ 0 ) and K ℓ 0 is defined similarly.

4.1.7

For any subfield F ′ ⊆ C which contains 𝐹 and s ∈ S ⁢ ( F ′ ) , by p s ∈ Mot AH ⁢ ( F ′ ) , we denote the submotive of h 2 ⁢ ( X s ) ⁢ ( 1 ) such that ω ? ⁢ ( p s ) = P ? , s for ? = B (when F ′ = C ), or dR , ét . Note that, for any σ ∈ Aut ⁢ ( C / F ) and s ∈ S ⁢ ( C ) , there is a natural isomorphism ( p s ) σ ≅ p σ ⁢ ( s ) in Mot AH ⁢ ( C ) , which we shall label as σ p , s ; moreover, the following diagrams tautologically commute (recall the notation in Section 2.1.3 and Notation 3.1.4):

4.1.8

For the rest of Section 4, we will put a standing assumption that 𝑆 is geometrically connected as an 𝐹-variety, so that S C = S ∘ .

Let K ⊆ G ⁢ ( A f ) be a neat compact open subgroup and assume that it is also admissible, so that μ b extends to a 𝖪-level structure [ μ ] on 𝖯 with [ μ ] b = K ⁢ ( μ b ⊗ A f ) . Now Lemma 3.1.7 guarantees that the restriction of [ μ ] to S C is ( P | S C ) -rational and is of type Ω. Therefore, Theorem 3.2.2 gives us a unique morphism ρ C : S C → Sh K ⁢ ( G ) C such that

(4.1) ( ρ C ) ∗ ⁢ ( V , [ η V ] ) ≅ ( P , [ μ ] ) | S C .

Note that, as 𝖪 is neat, the above isomorphism is unique; we denote its étale component by α ét ∘ : ρ C ∗ ⁢ ( V ét ) ⁢ → ∼ ⁢ P ét | S C and let α ℓ ∘ be its ℓ-adic component for a prime ℓ.

Theorem 4.1.9

In the notation above, assume either

p s ∈ Mot Ab ⁢ ( C ) for every s ∈ S C (e.g., when the family ( X / S , ξ ) | S C belongs to case (CM)), or
the family ( X / S , ξ ) | S C belongs to case (R+) = (R1) + (R2) and 𝖪 satisfies condition (♯) as defined in Section 4.1.6.

Then ρ C descends to a morphism ρ : S → Sh K ⁢ ( G ) F over 𝐹. Moreover, α ét ∘ descends to an isomorphism α ét : ρ ∗ ⁢ ( V ét ) ⁢ → ∼ ⁢ P ét over 𝑆.

Case (a) is easy: the diagrams in Section 4.1.7 readily imply that 𝖯 over 𝑆 is weakly AM, so that the conclusion follows from Theorem 3.2.5. In fact, we have that the de Rham component of the isomorphism (4.1) also descends to 𝑆. Note that case (a) in particular covers the case when ( X / S , ξ ) | S C belongs to case (CM) by Theorem 2.2.7 (d). The proof of case (b) is the content of Section 4.2 below. If ( X / S , ξ ) | S C belongs to case (R2^′) and it is not known that p s ∈ Mot Ab ⁢ ( C ) for every s ∈ S C , a slightly weaker version of the above theorem holds.

Theorem 4.1.10

Assume that the family ( X / S , ξ ) | S ∘ belongs to case (R2^′), Λ = Λ 0 , 𝖪 satisfies condition (♯), and for a prime ℓ 0 , K ℓ 0 is sufficiently small as defined in Section 4.1.6. Then ρ C : S C → Sh K ⁢ ( G ) C descends to a morphism ρ : S F ′ → Sh K ⁢ ( G ) F ′ for a finite extension F ′ / F in ℂ. Moreover, α ℓ 0 ∘ descends to an isomorphism α ℓ 0 : ρ ∗ ⁢ V ℓ 0 ⁢ → ∼ ⁢ P ℓ 0 | S F ′ , and ρ ∗ ⁢ V ℓ is étale locally isomorphic to P ℓ | S F ′ for every other ℓ in the sense of Definition 4.1.1.

Remark 4.1.11

In literature, to define period morphisms to orthogonal Shimura varieties, one usually keeps track of a trivialization of the determinants (cf. [40, Proposition 4.3]). Such a trivialization (i.e., an isometry det ( V ) ¯ ⁢ → ∼ ⁢ det ( P B ) in our notation) is implicit in the statement that (the restriction to S C of) [ μ ] is ( P | S C ) -rational as a ( G , V , K ) -level structure, because det ( V ) is 𝐺-invariant. More concretely, one obtains this trivialization by globalizing det ( μ b ) , using that π 1 ⁢ ( S C , b ) fixes det ( P B , b ) (cf. the proof of Lemma 3.1.7). However, we remark that if 𝑆 were not geometrically connected, we would not be able to show that [ μ ] is ( P | S C ) -rational over the connected components of S C other than S ∘ , unless we know det ( P B , b ) is spanned by an absolute Hodge class (e.g., in Theorem 4.1.9 (a)).

On the other hand, for the purpose of putting level structures, the lack of absolute Hodgeness of det ( P B , b ) is partially remedied by independence-of-ℓ type results on algebraic monodromy (e.g., [55, Lemma 3.2]; cf. [57, Corollary 5.9]), which implies that if, for some ℓ, det ( P B , b ) ⊗ Q ℓ is π 1 ét ⁢ ( S , b ) -invariant, then the same is true for all ℓ. In Lemma 4.1.4, we used Larsen–Pink’s result [38, Proposition 6.14] to achieve a similar effect. Later, in Section 4.2.4, this is used to overcome a similar difficulty: we do not know that the tensors which cut out Res E / Q ⁡ SO ⁢ ( V ) from Res E / Q ⁡ O ⁢ ( V ) are given by absolute Hodge tensors on P B , b via μ b . As far as we are aware of, this cannot be deduced from Theorem 2.2.7. However, we can put 𝒦-level structures on P ét in question and proceed.

4.2 Case (R+): Maximal monodromy

Lemma 4.2.1

It suffices to prove Theorem 4.1.9 when Λ = NS ⁢ ( X η ) Q .

Proof

Let Λ ⟂ be the orthogonal complement of Λ in Λ 0 : = NS ( X η ) Q and set

M = μ b − 1 ⁢ ( Λ ⟂ ) .

Then V = V 0 ⊕ M and we view G 0 as the stabilizer of 𝑀 in 𝐺. The image of π 1 ét ⁢ ( S , b ) in G ⁢ ( A f ) via μ b actually lies in G 0 ⁢ ( A f ) . Therefore, K 0 = K ∩ G 0 ⁢ ( A f ) satisfies condition (♯) for Λ 0 .

Let V 0 be the automorphic system on Sh 0 : = Sh K 0 ( G 0 ) given by V 0 ∈ Rep ⁢ ( G 0 ) , let [ η V 0 ] be the K 0 -level structure on V 0 , and let [ μ 0 ] be the K 0 -level structure on P 0 defined by μ 0 , b : = μ b | V 0 . As in the paragraph above Theorem 4.1.9, we obtain a morphism

ρ 0 , C : S C → Sh 0 , C

such that ( V 0 , [ η V 0 ] ) | S C ≅ ( P 0 , [ μ 0 ] ) | S C . Define α 0 , ét ∘ accordingly.

Consider the Shimura morphism j : Sh 0 → Sh : = Sh K ( G ) . Then we have natural identifications

j ∗ ⁢ ( V ) = V 0 ⊕ ( M ⊗ 1 Sh 0 ) and P = P 0 ⊕ ( Λ ⟂ ⊗ 1 S )

Moreover, the level structure [ η V 0 ] (resp. [ μ 0 ] ) refines j ∗ ⁢ ( [ η V ] ) (resp. [ μ ] ) in the sense of Definition 3.1.2 (d). Therefore, one easily checks that ( j C ∘ ρ 0 , C ) ∗ ⁢ ( V , [ η V ] ) ≅ ( P , [ μ ] ) | S C . By the uniqueness statement in Theorem 3.2.2, this implies that ρ C = j C ∘ ρ 0 , C .

Assume now that ρ 0 , C descends to ρ 0 over 𝐹, and α 0 , ét ∘ descends to α 0 , ét over 𝑆. Then ρ C descends to j F ∘ ρ 0 , and α ét ∘ descends to an isomorphism α 0 , ét ⊕ ( id Λ ⟂ ⊗ A ¯ f ) over 𝑆. ∎

4.2.2

In this section, we prove Theorem 4.1.9. By Lemma 4.2.1, we may assume that Λ = Λ 0 , so that V = V 0 and P = P 0 . Let E , G , V , G ♯ , K , K be as introduced in Section 4.1.6. In particular, 𝐸 is the endomorphism field of the Hodge structure P b , and 𝒱 is the 𝐸-bilinear lift of 𝑉, which carries an 𝐸-action via the isometry μ b : V ⁢ → ∼ ⁢ P B , b fixed in Section 4.1.5.

Note that, by Theorem 2.2.7, each e ∈ E , viewed as an element of

End ⁡ ( P B , b ) = End ⁡ ( ω B ⁢ ( p b ) ) ,

is absolute Hodge, so its image in End ⁡ ( P ét , b ) is stabilized by an open subgroup of π 1 ét ⁢ ( S , b ) . Since we assumed that Mon ⁢ ( P ℓ , b ) is connected for every ℓ, the 𝐸-action on P ét , b must already be π 1 ét ⁢ ( S , b ) -equivariant. This has the following consequence.

Lemma 4.2.3

The action of 𝐸 on P B , b extends to an action on 𝖯. If τ : E ↪ C is the embedding induced by the action of 𝐸 on Fil 1 ⁢ P dR , b , or equivalently the unique indefinite real place of 𝒱, then τ ⁢ ( E ) ⊆ F .

Proof

For the first statement, it is clear that the 𝐸-action on P B , b (resp. P ét , b ) extends (necessarily uniquely) to P B (resp. P ét ) because it is π 1 ⁢ ( S C , b ) (resp. π 1 ét ⁢ ( S , b ) )-equivariant. It remains to show that the 𝐸-action on P dR | S C , obtained via the Riemann–Hilbert correspondence, descends to an action on P dR . This follows from the fact that the de Rham realization of every e ∈ E , as an element of P dR , b ⊗ , descends to P dR , η ⊗ : since 𝑒 is absolute Hodge and its étale component is π 1 ét ⁢ ( η , b ) -invariant, its de Rham component descends to 𝜂 (cf. the argument for [33, (2.2.2)]).

Recall that, by S ⁢ ( C ) , we mean the set of 𝐹-linear morphisms Spec ⁡ ( C ) → S . The first statement implies that, for every s ∈ S ⁢ ( C ) , P B , s carries an action of 𝐸, which is self-adjoint by Theorem 2.2.2; moreover, for every σ ∈ Aut ⁢ ( C / F ) , the 𝜎-linear isomorphism

σ P dR , s : P dR , s ≅ P dR , σ ⁢ ( s )

of filtered vector spaces is 𝐸-equivariant (see Notation 3.1.4). Therefore, if we let τ s : E → R be the place through which 𝐸 acts on Fil 1 ⁢ P dR , s , then τ σ ⁢ ( s ) = σ ∘ τ s . Now we use that τ s can also be characterized as the unique real place of 𝐸 such that P B , s ⊗ τ s R is indefinite. Parallel transport implies that τ s is constant on S ⁢ ( C ) , which by assumption is connected. As τ = τ b and σ ⁢ ( b ) ∈ S ⁢ ( C ) , σ ∘ τ = τ for every σ ∈ Aut ⁢ ( C / F ) . This implies that τ ⁢ ( E ) ⊆ F . ∎

4.2.4

Below, we shall view 𝐸 as a subfield of 𝐹 (and hence of ℂ) via 𝜏 as above and drop 𝜏 from the notation. Now we recall the discussion in Section 3.3.7. Let Ω V ⊆ Ω be the Hermitian symmetric subdomain { w ∈ P ⁢ ( V ⊗ E C ) ∣ ⟨ w , w ̄ ⟩ > 0 , ⟨ w , w ⟩ = 0 } . Then ( G , Ω V ) is a Shimura subdatum of ( G , Ω ) with reflex field 𝐸. Therefore, there is a Shimura morphism i C : Sh K ⁢ ( G ) C → Sh K ⁢ ( G ) C which descends to i : Sh K ⁢ ( G ) → Sh K ⁢ ( G ) E over 𝐸. Let V ̃ be the automorphic system on Sh K ⁢ ( G ) defined by 𝑉 and let [ η V ] be its tautological 𝒦-structure. Recall that V ̃ is identified with i ∗ ⁢ V , and [ η V ] refines i ∗ ⁢ ( [ η V ] ) .

Consider the situation in Section 4.1.8. Recall that we assumed that Mon ⁢ ( P ℓ , b ) is connected for every ℓ. Because the centralizer of the 𝐸-action in O ⁢ ( V ) can be identified with Res E / Q ⁡ O ⁢ ( V ) , which contains 𝒢 as the identity component, the monodromy action of π 1 ét ⁢ ( S , b ) must take values in K : = K ∩ G ( A f ) via μ b . Therefore, there exists a ( G , V , K ) -level structure [ μ ̃ ] on P ét such that [ μ ̃ ] b = K ⋅ ( μ b ⊗ A f ) . As the Hodge structure on P B , b is defined by a point on Ω V via μ b , by Lemma 3.1.7, the restriction of [ μ ̃ ] to S C is ( P | S C ) -rational and of type Ω V . One easily checks the following.

Lemma 4.2.5

Let ϱ C : S C → Sh K ⁢ ( G ) C be the unique morphism such that

(4.2) ( ϱ C ) ∗ ⁢ ( V ̃ , [ η V ] ) ≅ ( P , [ μ ̃ ] ) | S C

given by Theorem 3.2.2. Then ρ C = i C ∘ ϱ C .

Hence we reduce Theorem 4.1.9 to the following.

Theorem 4.2.6

Under the hypothesis of Theorem 4.1.9 and notation above, the morphism ϱ C descends to an 𝐹-morphism ϱ : S → Sh K ⁢ ( G ) F ; moreover, the étale component α ét ∘ : ϱ C ∗ ⁢ V ̃ ét ⁢ → ∼ ⁢ P ét | S C of (4.2) descends to an isomorphism of α ét : ϱ ∗ ⁢ V ̃ ét ⁢ → ∼ ⁢ P ét over 𝑆.

Note that α ét ∘ above agrees with the one in Theorem 4.1.9 because V ̃ ét = i ∗ ⁢ V ét .

Below, for any ℚ-linear Tannakian category 𝒞, we write N ⁢ ( − ) for the functor

Mod E ⁢ ( C ) → C

which sends every M ∈ Mod E ⁢ ( C ) to Nm E / Q ⁢ ( M ) ⊗ Q det ( M ( Q ) ) (cf. Section 2.1.4). Recall that, in Section 3.3.5, we explained how to apply N ⁢ ( − ) to a 𝒦-level structure on a system of realizations.

4.2.7

We first treat the case when m : = dim E V is odd. In this case,

N : = N ( V ) ∈ Rep ( G )

is faithful. Let 𝖭 be the automorphic system on Sh K ⁢ ( G ) associated to 𝑁. Let [ η N ] be the tautological 𝒦-level structure on 𝖭. Then, by Section 3.3.7, there is a unique isomorphism

(4.3) ( N ⁢ ( V ̃ ) , N ⁢ ( η V ) ) ≅ ( N , [ η N ] ) .

As remarked in Remark 3.3.4, one should read Section 3.3.7 with ( H , C , Ω H ) = ( G , K , Ω V ) .

Lemma 4.2.8

N ⁢ ( P ) ∈ R ⁢ ( S ) is weakly AM in the sense of Definition 3.1.5.

Proof

As the formation of N ⁢ ( − ) on Mot AH ⁢ ( C ) ( E ) commutes with cohomological realizations, for every s ∈ S ⁢ ( C ) , we have N ⁢ ( ( P B , s , P dR , s ) ) = ω Hdg ⁢ ( N ⁢ ( p s ) ) . By Theorem 2.2.7 (a), N ⁢ ( p s ) ∈ Mot Ab ⁢ ( C ) . Therefore, in Definition 3.1.5, we may take M = N ⁢ ( p s ) , so that

M σ = ( N ⁢ ( p s ) ) σ = N ⁢ ( ( p s ) σ ) .

By applying N ⁢ ( − ) to the objects in the diagram in Section 4.1.7, one checks that the diagrams in Definition 3.1.5 commute for N ⁢ ( P ) . ∎

Lemma 4.2.9

There exists a unique morphism ρ N : S → Sh K ⁢ ( G ) F such that

(4.4) ρ N ∗ ⁢ ( N , [ η N ] ) ≅ ( N ⁢ ( P ) , N ⁢ ( [ μ ̃ ] ) ) .

Moreover, ϱ C = ρ N | S C .

Proof

One easily checks from Section 3.3.5 that N ⁢ ( [ μ ̃ ] ) = K ⋅ ( N ⁢ ( μ b ) ⊗ A f ) , so by Lemma 3.1.7, N ⁢ ( [ μ ̃ ] ) is N ⁢ ( P ) -rational. Moreover, as [ μ ̃ ] is of type Ω V , so is N ⁢ ( [ μ ̃ ] ) . As N ⁢ ( P ) is weakly AM, Theorem 3.2.5 gives the ρ N for which (4.4) holds. On the other hand, by applying N ⁢ ( − ) to (4.2), we see that

(4.5) ( ϱ C ) ∗ ⁢ ( N , [ η N ] ) ≅ ( N ⁢ ( P ) , N ⁢ ( [ μ ̃ ] ) ) | S C .

By the uniqueness statement in Theorem 3.2.2, this implies that ϱ C = ρ N | S C . ∎

Proof of Theorem 4.2.6 for 𝑚 odd

To affirm the first statement, set ϱ : = ρ N . The second statement now follows from Lemma 3.3.6. Indeed, comparing (4.3), (4.4), and (4.5), we see that N ⁢ ( α ét ∘ ) descends to 𝑆. However, as dim E V is odd, Z ⁢ ( V ) = 1 (see Notation 2.1.5). By Lemma 3.3.6, α ét ∘ descends to 𝑆. ∎

4.2.10

Recall that, in Section 4.1.6, V ♯ = V ⊕ E for ℰ defined in Notation 2.1.5. If m = dim E V is even, we let V ♯ play the role of 𝒱 in the above proof. Recall that, in Theorem 4.1.9, we assumed that 𝖪 satisfies condition (♯) and we are currently in situation V = V 0 , so that K = K ∩ G ⁢ ( A f ) ⊆ U for some neat compact open U ⊆ G ♯ ⁢ ( A f ) . Define the Hermitian symmetric domain Ω V ♯ with 𝒱 replaced by V ♯ in Ω V . Then we obtain an embedding of Shimura data ( G , Ω V ) ↪ ( G ♯ , Ω V ♯ ) . By [12, (1.15)], for some U ′ ⊇ K , the Shimura morphism Sh K ⁢ ( G ) → Sh U ′ ⁢ ( G ♯ ) is an embedding. Replacing 𝒰 by U ∩ U ′ if necessary, we may assume that the Shimura morphism Sh K ⁢ ( G ) → Sh U ⁢ ( G ♯ ) is also an embedding. Below, we write this embedding simply as j : Sh K → Sh U . Let 𝖶 be the automorphic system of realizations on Sh U given by W : = ( V ♯ ) ( Q ) ∈ Rep ( G ♯ ) and let [ η W ] be the tautological 𝒰-level structure on 𝖶.

The reader should now apply the discussion in Section 3.3.7 with ( G , Ω V ) replaced by ( G ♯ , Ω V ♯ ) , V ̃ replaced by 𝖶, ( H , C , Ω H ) = ( G ♯ , U , Ω V ♯ ) , and π = id (cf. Remark 3.3.4). In particular, 𝖶 is equipped with a natural 𝐸-action. This time, we set N = N ⁢ ( V ♯ ) ∈ Rep ⁢ ( G ♯ ) . Let 𝖭 be the automorphic system on Sh U given by 𝑁 and let [ η N ] be the tautological 𝒰-level structure. Note that N ∈ Rep ⁢ ( G ♯ ) is faithful as dim E V ♯ is odd. Now Section 3.3.7 tells us that

( N , [ η N ] ) = ( N ⁢ ( W ) , N ⁢ ( [ η W ] ) )

It is not hard to see that the restriction of 𝖶 to Sh K is naturally identified with V ̃ ⊕ ( E ⊗ 1 Sh K ) . Correspondingly, we set Q : = P ⊕ ( E ⊗ 1 S ) and define ν b : W ⁢ → ∼ ⁢ Q B , b by μ b ⊕ id E . Then ν b defines a 𝒰-level structure [ ν ] with [ ν ] b = U ⋅ ( ν b ⊗ A f ) . Define β C : = j C ∘ ϱ C . Then we have

(4.6) ( β C ) ∗ ⁢ ( W , [ η W ] ) ≅ ( Q , [ ν ] ) | S C .

Lemma 4.2.11

There exists a unique morphism β N : S → ( Sh U ) F such that

(4.7) β N ∗ ⁢ ( N , [ η N ] ) ≅ ( N ⁢ ( Q ) , N ⁢ ( [ ν ] ) ) .

Moreover, β C = β N | S C .

Proof

By Theorem 2.2.7 (b) and a slight variant of the argument for Lemma 4.2.8, N ⁢ ( Q ) is weakly AM. As N ⁢ ( [ ν ] ) b = U ⋅ ( N ⁢ ( μ b ) ⊗ A f ) , Lemma 3.1.7 says that the ( G ♯ , N , U ) -level structure N ⁢ ( [ ν ] ) is N ⁢ ( Q ) -rational. As [ μ ̃ ] is of type Ω V , [ ν ] is of type Ω V ♯ , so that N ⁢ ( [ ν ] ) is also of type Ω V ♯ . Applying Theorem 3.2.5 to the faithful representation 𝑁 of G ♯ , we obtain the desired map β N such that (4.7) holds. By the uniqueness statement in Theorem 3.2.2, to show β C = β N | S C , it suffices to observe that

β C ∗ ⁢ ( N ⁢ ( W ) , N ⁢ ( [ η W ] ) ) ≅ ( N ⁢ ( Q ) , N ⁢ ( [ ν ] ) ) | S C .

One checks this by applying N ⁢ ( − ) to (4.6). ∎

Proof of Theorem 4.2.6 for 𝑚 even

The above implies that β C descends to β N over 𝐹. Recall that β C = j C ∘ ϱ C and j C is an embedding. As the actions on ℂ-points of both β C and j C are Aut ⁢ ( C / F ) -equivariant, the same is true for ϱ C , so that ϱ C descends to a morphism 𝜚 over 𝐹 with β N = j F ∘ ϱ .

Let λ ét ∘ : ( W ét , [ η W ] ) | S C ≅ ( Q ét , [ ν ] ) | S C be the étale component of (4.6). Then N ⁢ ( λ ét ∘ ) is the étale component of (4.7) restricted to S C . Therefore, N ⁢ ( λ ét ∘ ) descends to 𝑆. Applying Lemma 3.3.6 to V ♯ , we have that λ ét ∘ descends to an isomorphism λ ét : β N ∗ ⁢ W ét ≅ Q ét over 𝑆. Note the decompositions

β N ∗ ⁢ W = ϱ ∗ ⁢ V ̃ ⊕ ( E ⊗ 1 S ) and Q = P ⊕ ( E ⊗ 1 S ) .

Since λ ét ∘ respects these decompositions over S C by construction, its descent λ ét over 𝑆 must also respect these decompositions, which are defined over 𝑆. Hence λ ét restricts to the sought after α ét : ϱ ∗ ⁢ V ̃ ét ⁢ → ∼ ⁢ P ét in Theorem 4.2.6. ∎

4.3 Case (R2^′): Non-maximal monodromy

By Proposition 2.2.5, we expect this case to be rare in practice. Readers who are not particularly interested in this case might skip to the next section.

Lemma 4.3.1

Let k ⊆ C be a subfield and let A , B be 𝑘-varieties with 𝐴 being geometrically connected. Suppose that there is a morphism f : A C → B C over ℂ, and an étale morphism g : B → C over 𝑘 such that, for some h : A → C , g C ∘ f = h C . Then 𝑓 descends to a subfield k ′ of ℂ which is finite over 𝑘 such that g k ′ ∘ f = h k ′ .

Proof

We assume without loss of generality that 𝑘 is algebraically closed. The graph Γ f : A C → ( A × C B ) C of 𝑓 (as a morphism between C C -schemes) defines a section of the étale morphism ( g × C h ) C : ( A × C B ) C → A C . Hence Γ f maps A C isomorphically onto a connected component D C of ( A × C B ) C . Since 𝑘 is algebraically closed, D C comes from an extension of scalars of a connected component D ⊂ ( A × C B ) . As the natural projection D → A is defined over 𝑘 and its base change to ℂ is the inverse of Γ f , we must have that Γ f is also defined over 𝑘, and hence so is 𝑓. ∎

Below, for any ℚ-linear Tannakian category 𝒞, we write N ⁢ ( − ) for the functor

Nm E / Q : Mod E ⁢ ( C ) → C

(cf. Section 2.1.4).

Proof of Theorem 4.1.10

As before, set K : = K ∩ G ( A f ) and let

i : Sh K ⁢ ( G ) → Sh K ⁢ ( G ) E

be the Shimura morphism. Our discussions in Section 4.2.2 up to Lemma 4.2.5 apply without any change in the (R2^′) case, so that ρ C factors through a morphism ϱ C : S C → Sh K ⁢ ( G ) C such that

(4.8) ϱ C ∗ ⁢ ( V ̃ , [ η V ] ) ≅ ( P , [ μ ̃ ] ) | S C ,

and we still have E ⊆ F . We first show that ϱ C descends to a morphism ϱ : S F ′ → Sh K ⁢ ( G ) F ′ for some finite extension F ′ / F in ℂ.

Set N : = N ( V ) . Then 𝑁 is a faithful representation of H = H ⁢ ( V ) (Notation 2.1.5). As we assumed in Theorem 4.1.10 that 𝖪 satisfies condition (♯), there exists a neat C ⊆ H ⁢ ( A f ) such that the image of 𝒦 lies in 𝒞. Now recall our discussion in Section 3.3.7 and the notation therein. Let π : Sh K ⁢ ( G ) → Sh C ⁢ ( H ) be the natural Shimura morphism over 𝐸. Let N ⁢ ( [ μ ̃ ] ) denote the 𝒞-level structure on N ⁢ ( P ét ) such that N ⁢ ( [ μ ̃ ] ) b = C ⋅ ( N ⁢ ( μ ̃ b ) ⊗ A f ) . By applying N ⁢ ( − ) to the diagrams in Section 4.1.7, Theorem 2.2.7 (c) implies that N ⁢ ( P ) is weakly AM. One checks using Lemma 3.1.7 that N ⁢ ( μ ̃ ) is N ⁢ ( P ) -rational, and is of type Ω H . Then, by Theorem 3.2.5, we obtain a morphism ρ N : S → Sh C ⁢ ( H ) F such that

(4.9) ( ρ N ) ∗ ⁢ ( N , [ η N ] ) ≅ ( N ⁢ ( P ) , N ⁢ ( [ μ ̃ ] ) ) .

By applying N ⁢ ( − ) to (4.8) and comparing with (3.2) in Section 3.3.7, for β C : = π C ∘ ϱ C , we obtain

β C ∗ ⁢ ( N , [ η N ] ) = ϱ C ∗ ⁢ π C ∗ ⁢ ( N , [ η N ] ) ≅ ϱ C ∗ ⁢ ( N ⁢ ( V ̃ ) , N ⁢ ( [ η V ] ) ) ≅ ( N ⁢ ( P ) , N ⁢ ( [ μ ̃ ] ) ) | S C .

Therefore, by the uniqueness statement in Theorem 3.2.2, β C = ρ N | S C , i.e., β C is defined over 𝐹. As 𝜋 is étale and is defined over E ⊆ F , by Lemma 4.3.1, ϱ C descends to a morphism 𝜚 over some finite extension F ′ / F in ℂ such that ( ρ N ) F ′ = π F ′ ∘ ϱ .

The above gives the first statement of Theorem 4.1.10 and we now turn to the second. Recall that we defined α ét ∘ : ρ C ∗ ⁢ ( V ét , [ η V ] ) ⁢ → ∼ ⁢ ( P , [ μ ] ) | S C in Section 4.1.8. Since V ̃ ét = i ∗ ⁢ V ét and ρ C = i C ∘ ϱ C , we may alternatively view α ét ∘ as the étale component of (4.8), i.e., an isomorphism ϱ C ∗ ⁢ V ̃ ét ⁢ → ∼ ⁢ P ét | S C , which sends ϱ C ∗ ⁢ [ η V ] to [ μ ̃ ] . As ( ρ N ) F ′ = π F ′ ∘ ϱ , (4.9) gives us an isomorphism

ϱ ∗ ⁢ ( N ⁢ ( V ̃ ét ) , N ⁢ ( [ η V ] ) ) = ϱ ∗ ⁢ π ∗ ⁢ ( N ét , [ η N ] ) ≅ ( N ⁢ ( P ét ) , N ⁢ ( [ μ ̃ ] ) ) | S F ′

whose restriction to S C is N ⁢ ( α ét ∘ ) . This implies that N ⁢ ( α ét ∘ ) descends to S F ′ . As we assumed that K ℓ 0 ∩ Z ⁢ ( Q ℓ 0 ) = 1 , Lemma 3.3.6 tells us that the ℓ 0 -adic component

α ℓ 0 ∘ : V ̃ ℓ 0 | S C ⁢ → ∼ ⁢ P ℓ 0 | S C

descends to S F ′ . For every other ℓ, K ℓ still contains an open subgroup K ℓ ′ such that

K ℓ ′ ∩ Z ⁢ ( Q ℓ ) = 1 .

Hence Lemma 3.3.6 implies that ϱ ∗ ⁢ V ̃ ℓ is étale-locally isomorphic to P ℓ | S F ′ . ∎

Remark 4.3.2

We remark that Theorem 4.1.10 is slightly weaker than Theorem 4.1.9 (e.g., one cannot descend ρ C to 𝐹 but only to some finite extension) fundamentally because the representation G → GL ⁢ ( N ) is not faithful, but has a finite kernel. In the (R2) case, this was avoided because we worked with V ♯ instead.

5 Proof of Theorem B

5.1 A specialization lemma for monodromy

Definition 5.1.1

Let 𝑆 be a Noetherian integral normal scheme. Let ℓ ∈ O S × be a prime and W ℓ an étale Q ℓ -local system. We denote by λ ⁢ ( W ℓ ) the dimension dim lim → U ⁡ W ℓ , s U , where 𝑠 is a geometric point on 𝑆 and 𝑈 runs through open subgroups of π 1 ét ⁢ ( S , s ) .

It is clear that the definition is independent of the choice of 𝑠.

Definition 5.1.2

Let 𝑇 be a Noetherian base scheme and let 𝑆 be a smooth 𝑇-scheme of finite type.

Let ℓ ∈ O T × be a prime and W ℓ an étale Q ℓ -local system. We say that W ℓ has constant λ geo if there exists a number 𝜆 such that, for every geometric point t → T and every connected component S ∘ of S t , λ ⁢ ( W ℓ | S ∘ ) = λ . When this condition is satisfied, write λ geo ⁢ ( W ℓ ) for 𝜆.
We say that S ̄ is a good relative compactification of 𝑆 if S ̄ is a smooth proper 𝑇-scheme and there exists a relative normal crossing divisor 𝐷 of S ̄ such that S = S ̄ − D .

Lemma 5.1.3

Let 𝑇 be a DVR with special point 𝑡 and generic point 𝜂. Assume that char ⁡ k ⁢ ( η ) = 0 and ℓ ∈ O T × . Let S → T be a smooth morphism of finite type with 𝑆 being connected. Let W ℓ be a Q ℓ -local system over 𝑆. If 𝑆 admits a good relative compactification S ̄ over 𝑇, then W ℓ has constant λ geo over 𝑇.

Proof

Let T ̃ be the strict Henselianization of 𝑇 and let t ̃ and η ̃ be the special and generic point of T ̃ . Let S ′ be a connected component of S T ̃ . Then [66, Tag 055J] tells us that S η ̃ ′ is connected. As S η ̃ ′ necessarily contains a k ⁢ ( η ̃ ) -rational point, S η ̃ ′ is geometrically connected. Now, by applying [66, Tag 0E0N] to S ̄ , S η ̃ and S t ̃ have the same number of geometric connected components, so S t ̃ ′ must be connected. To prove the lemma, we may replace 𝑇 by T ̃ and 𝑆 by S ′ , so that S t and S η are both geometrically connected. Let η ̄ be the geometric point over 𝜂 defined by a chosen algebraic closure of k ⁢ ( η ) .

Choose a section σ : T → S and set a = σ ⁢ ( t ) , b = σ ⁢ ( η ̄ ) . Note that 𝜎 provides an étale path between 𝑎 and 𝑏, through which we identify W ℓ , a with W ℓ , b and π 1 ét ⁢ ( S , a ) with π 1 ét ⁢ ( S , b ) . Let ρ : π 1 ét ⁢ ( S , a ) → GL ⁢ ( W ℓ , a ) be the monodromy representation. For any group 𝐺 with a morphism G → π 1 ét ⁢ ( S , a ) implicitly understood, write ρ ⁢ ( G ) ∘ for the identity component of the Zariski closure of the image of 𝐺 in GL ⁢ ( W ℓ , a ) . Clearly, ρ ⁢ ( G ) ∘ remains unchanged if we replace 𝐺 by a finite index subgroup. It suffices to show that ρ ⁢ ( π 1 ét ⁢ ( S t , a ) ) ∘ = ρ ⁢ ( π 1 ét ⁢ ( S η ̄ , b ) ) ∘ , i.e., Mon ∘ ⁢ ( W ℓ | S t , a ) = Mon ∘ ⁢ ( W ℓ | S η ̄ , b ) in the notation in Definition 4.1.1.

Take a sequence F n of locally constant free Z / ℓ n ⁢ Z -modules over 𝑆 such that

W ℓ ≅ ( lim ← n ⁡ F n ) ⊗ Q ℓ .

As char ⁡ k ⁢ ( η ) = 0 , each F n is tamely ramified over S ̄ by Abhyankar’s lemma [23, XIII, Appendix, Proposition 5.5]^[8]. Let us use a superscript “𝑡” to indicate tame fundamental group. Then we know that ρ ⁢ ( π 1 ét ⁢ ( S ? , a ) ) ∘ = ρ ⁢ ( π 1 ét ⁢ ( S ? , a ) t ) ∘ for ? = ∅ , t . By [23, Exposition XIII 2.10], the natural map π 1 ét ⁢ ( S t , a ) t → π 1 ét ⁢ ( S , a ) t is an isomorphism, so it remains to show that

ρ ⁢ ( π 1 ét ⁢ ( S η ̄ , b ) ) ∘ = ρ ⁢ ( π 1 ét ⁢ ( S , b ) t ) ∘ .

The section σ ⁢ ( η ) induces an isomorphism π 1 ét ⁢ ( S , b ) = π 1 ét ⁢ ( S η ̄ , b ) ⋊ Gal k ⁢ ( η ) . As σ ∗ ⁢ ( W ℓ ) is necessarily trivial, the subgroup Gal k ⁢ ( η ) ⊆ π 1 ét ⁢ ( S , b ) acts trivially on W ℓ , b . Therefore,

ρ ⁢ ( π 1 ét ⁢ ( S η ̄ , b ) ) ∘ = ρ ⁢ ( π 1 ét ⁢ ( S η , b ) ) ∘ = ρ ⁢ ( π 1 ét ⁢ ( S , b ) t ) ,

as desired. Note that the second equality follows from the simple fact that π 1 ét ⁢ ( S η , b ) maps surjectively to π 1 ét ⁢ ( S , b ) , and π 1 ét ⁢ ( S , b ) t is a quotient of π 1 ét ⁢ ( S , b ) . ∎

Proposition 5.1.4

Let 𝑆 be a connected smooth ℂ-variety and let f : X → S be a ♡ -family. Let 𝑠 be any Hodge-generic point on 𝑆. Then, for any prime ℓ and V ℓ : = R 2 f ∗ Q ℓ ( 1 ) , dim NS ⁢ ( X s ) Q = λ ⁢ ( V ℓ ) .

Proof

Up to replacing 𝑆 by a connected étale cover, assume that M ℓ : = Mon ( V ℓ , s ) is connected (see Definition 4.1.1). Set ρ : = NS ( X s ) Q . By Lemma 4.1.2, ρ = NS ⁢ ( X η ) Q . As M ℓ is unchanged if we replace 𝑆 by a further connected étale cover, dim V ℓ , s M ℓ = λ ⁢ ( V ℓ ) . Let V = ( V B , V dR ) be the VHS R 2 ⁢ f ∗ ⁢ Q ⁢ ( 1 ) . Then 𝖵 splits into 1 S ⊕ ρ ⊕ P for some VHS 𝖯 such that P B , s ( 0 , 0 ) = 0 . It suffices to argue that P ℓ , s M ℓ = 0 , where P ℓ = P B ⊗ Q ℓ .

Set M : = Mon ( V B , s ) (see Definition 2.2.3). Then we have M ℓ = M ⊗ Q ℓ . This implies that V ℓ , s M ℓ = V B , s M ⊗ Q ℓ , so we reduce to showing that P B , s M = 0 . We recall that Deligne’s theorem of the fixed part [11, (4.1.2)] says that the subspace P B , s M has a Hodge structure which is respected by the embedding P B , s M ↪ P B , s . Since the Hodge structure on P B , s is irreducible [26, §3, Lemma 2.7], P B , s M is either 0 or P B , s . But it cannot be P B , s because M ≠ 1 by our assumption that X / S is a ♡ -family. ∎

5.2 An effective theorem

First, we extend Set-up 4.1.3 and the set-up in Section 4.1.5 to a family over Z ( p ) when F = Q .

Set-up 5.2.1

Let 𝖬 be a connected separated scheme over Z ( p ) which is smooth and of finite type for some prime p > 2 . Let ( f : X → M ) be a smooth projective morphism of relative dimension 𝑑 such that X | M Q is a ♡ -family. Let 𝜂 be the generic point of 𝖬. Let 𝝃 be a relatively ample line bundle on X / M , which endows R 2 ⁢ f ∗ ⁢ A f p ⁢ ( 1 ) and R 2 ⁢ f Q ⁣ ∗ ⁢ Z p ⁢ ( 1 ) with a symmetric bilinear pairing. Let Λ ⊆ Λ 0 : = NS ( X η ) Q be a subspace which contains the class of ξ η . Recall that, by Lemma 4.1.2, Pic ⁡ ( X η ) = Pic ⁡ ( X ) / Pic ⁡ ( M ) . By choosing a section Λ 0 ↪ Pic ( X η ) Q , we obtain an embedding

Λ ¯ 0 → R 2 ⁢ f ∗ ⁢ A f p ⁢ ( 1 ) ,

and for every field 𝑘 and s ∈ M ⁢ ( k ) , Λ 0 (and hence Λ) is naturally a subspace of NS ⁢ ( X s ) Q . We write PNS ⁢ ( X s ) Q for the orthogonal complement of Λ in NS ⁢ ( X s ) Q . Note that these definitions are independent of the section Λ 0 ↪ Pic ( X η ) Q chosen. Choose a base point b ∈ M ⁢ ( C ) lying above 𝜂 and let the connected component of M C which contains 𝑏 be M ∘ .

We assume that Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q 2 , b ) is connected. Now apply Set-up 4.1.3 and the set-up in Section 4.1.5 with S = M Q and S ∘ = M ∘ and define a system of realizations

( P B , P dR , P ét ) ∈ R ⁢ ( M Q ) ;

moreover, we fix μ b : V ⁢ → ∼ ⁢ P B , b and define the Shimura datum ( G , Ω ) . Let ( R 2 ⁢ f Q ⁣ ∗ ⁢ Z p ) tf be the image of R 2 ⁢ f Q ⁣ ∗ ⁢ Z p in R 2 ⁢ f Q ⁣ ∗ ⁢ Q p and define ( R 2 ⁢ f C ⁣ ∗ ⁢ Z ( p ) ) tf similarly ( tf is short for “torsion-free”). Let P B : = P B ∩ ( R 2 f C ⁣ ∗ Z ( p ) ( 1 ) ) tf and P p : = P p ∩ ( R 2 f Q ⁣ ∗ Z p ( 1 ) ) tf . For every ℓ ≠ p , let P ℓ be the orthogonal complement of Λ in R 2 ⁢ f ∗ ⁢ Q ℓ ⁢ ( 1 ) , so that P ℓ over M Q extends to P ℓ over 𝖬. If 𝑘 is a perfect field of characteristic 𝑝 and W : = W ( k ) , for every point t ∈ M ⁢ ( k ) , ξ t defines a pairing on the F-isocrystal H cris 2 ⁢ ( X t / W ) ⁢ [ 1 / p ] and we write P cris , t ⁢ [ 1 / p ] for the orthogonal complement of the classes in Λ ⊆ NS ⁢ ( X t ) Q . Assume that the Z ( p ) -pairing on L ( p ) : = μ b − 1 ( P B , b ) is self-dual. We abusively write the reductive Z ( p ) -group SO ⁢ ( L ( p ) ) also as 𝐺.

Under the above set-up, we define the following.

Definition 5.2.2

Let K ⊆ G ⁢ ( A f ) be a neat compact open subgroup of the form K p ⁢ K p for K p = G ⁢ ( Z p ) and K p ⊆ G ⁢ ( A f p ) . Let S K ⁢ ( G ) denote the integral model of Sh K ⁢ ( G ) over Z ( p ) . Let ℓ 0 ≠ p be a prime. We say that a morphism ρ : M → S K ⁢ ( G ) is an ℓ 0 -admissible period morphism if (recall the notation in Section 3.4.2)

there exists an isometry α B ∘ : ρ C ∗ ⁢ L B | M ∘ ⁢ → ∼ ⁢ P B | M ∘ compatible with the Hodge filtrations (i.e., it induces an isomorphism ( ρ C ∗ ⁢ V ) | M ∘ ⁢ → ∼ ⁢ P | M ∘ of VHS over M ∘ );
there is an isometry α ℓ 0 : ρ ∗ ⁢ L ℓ 0 ⁢ → ∼ ⁢ P ℓ 0 whose restriction to M ∘ agrees with α B ∘ ⊗ Q ℓ 0 ;
for every ℓ ≠ p , ρ ∗ ⁢ L ℓ is étale-locally isomorphic to P ℓ over 𝖬;
ρ Q ∗ ⁢ L p is étale locally isomorphic to P p over M Q .

Note that the isomorphism α ℓ 0 is unique if it exists. If (b) is satisfied for every prime ℓ ≠ p , then we simply say that 𝜌 is admissible.

Theorem 5.2.3

Consider Set-up 5.2.1. Assume that, for some prime ℓ 0 ≠ p ,

P ℓ 0 has constant λ geo over Z ( p ) as defined in Definition 5.1.2, and
there is an ℓ 0 -admissible period morphism ρ : M → S K ⁢ ( G ) for some 𝖪 as in Definition 5.2.2.

Then, for every 𝑘 which is finitely generated over F p and t : Spec ⁡ ( k ) → M , the fiber X t satisfies the Tate conjecture for divisors.

Proof

Define G ̃ : = CSpin ( L ) and K p : = G ̃ ( Z p ) as in Section 3.4. Choose a compact open subgroup K p ⊆ G ̃ ⁢ ( A f p ) whose image is contained in K p such that K : = K p K p is neat. Up to replacing 𝖬 by a further connected étale cover, let us assume that 𝜌 can be lifted to a morphism M → S : = S K ( G ̃ ) . Below, we shall use 𝜌 to denote this lift. Recall the definition of special endomorphisms in Definition 3.4.1. Under these preparations, we have the following proposition.

Proposition 5.2.4

For every algebraically closed field 𝜅 and geometric point

s : Spec ⁡ ( κ ) → M ,

there is an isomorphism θ s : L ⁢ ( A ρ ⁢ ( s ) ) Q ⁢ → ∼ ⁢ PNS ⁢ ( X s ) Q such that the diagram

(5.1)

commutes, where the vertical arrows are cycle class maps.

Now we prove Theorem 5.2.3 assuming the proposition above. For any 𝑘 (not necessarily finite) and 𝑡 in Theorem 5.2.3, we claim that

(5.2) L ⁢ ( A t ) Q ⊗ Q ℓ = L ℓ , t ̄ Gal ⁢ ( t ̄ / t )

If 𝑘 is finite, the claim is given by [40, Theorem 6.4]. We remark that the ℓ-independence assumption in [40, (6.2)] can now be deduced from [34, Corollary 2.3.1]. When 𝑘 is not finite, the claim follows from the proof of [40, Corollary 6.11]. Indeed, we may assume that 𝑘 is the fraction field of some smooth and geometrically connected variety 𝑇 over a finite extension of F p , and 𝑡 extends to a morphism T → M . Let us choose a closed point t 0 on 𝑇, which we also view as a k ⁢ ( t 0 ) -valued point on 𝖬. Choose a geometric point t ̄ 0 over t 0 , and let 𝛾 be an étale path connecting t ̄ and t ̄ 0 . By [19, I, Proposition 2.7], End ⁡ ( A T ) = End ⁡ ( A t ) . This gives us a specialization morphism L ⁢ ( A t ) → L ⁢ ( A t 0 ) which fits into a commutative diagram below:

(recall the notation in Section 3.4). We claim that all vertical squares are Cartesian. For the squares on the left and right, this is clear by Definition 3.4.1. For the square at the front, this is obtained by applying (5.2) to t 0 . Hence the remaining diagram at the back must also be Cartesian. Therefore, the surjectivity of End ( A t ) ⊗ Q ℓ → End ( H ℓ , t ̄ ) Gal ⁢ ( t ̄ / t ) (see [64]) implies the surjectivity of L ⁢ ( A t ) ⊗ Q ℓ → L ℓ , t ̄ Gal ⁢ ( t ̄ / t ) . Hence we have affirmed (5.2) for 𝑡.

Finally, combining (5.1), (5.2), and Definition 5.2.2 (c), we have

dim PNS ⁢ ( X t ̄ ) Q = λ ⁢ ( t ∗ ⁢ P ℓ 0 ) = λ ⁢ ( t ∗ ⁢ L ℓ 0 ) = λ ⁢ ( t ∗ ⁢ L ℓ ) = λ ⁢ ( t ∗ ⁢ P ℓ )

This implies the Tate conjecture in codimension 1 for X t . ∎

Now we prove Proposition 5.2.4, which is the key geometric input to Theorem 5.2.3.

Proof

We first prove the statement when char ⁡ κ = 0 . Without loss of generality, we may assume that 𝜅 can be embedded to ℂ; moreover, as Aut ⁢ ( C ) acts transitively on the set of connected components of M C , we may choose an embedding such that the resulting ℂ-point lies on the distinguished component M ∘ (defined in Set-up 5.2.1) of 𝜌. To prove the statement, we may replace 𝑠 by this ℂ-point. Then the statement follows from Hodge theory. Indeed, we obtain a commutative diagram^[9]

The vertical maps are again cycle class maps, but this time they are isomorphisms. For the arrow on the right, we are applying the Lefschetz ( 1 , 1 ) -theorem. Since α ℓ 0 , s = α B , s ∘ ⊗ Q ℓ 0 by assumption, we obtain (5.1).

Now we assume that char ⁡ κ = p . Set W = W ⁢ ( κ ) . We shall construct θ s by considering characteristic 0 liftings of 𝑠. Let S ̂ s be the formal completion of S K ⁢ ( G ̃ ) W at ρ ⁢ ( s ) . For a special endomorphism ζ ∈ L ⁢ ( A s ) , consider the following functor:

(5.3) Def S ⁢ ( ζ , s ) : R ↦ { s ̃ ∈ S ̂ s ⁢ ( R ) ∣ ζ ⁢ deforms to ⁢ L ⁢ ( A s ̃ ) } ,

where 𝑅 runs through all Artin 𝑊-algebras. By [41, §5.14], Def S ⁢ ( ζ , s ) is represented by a closed formal subscheme of S ̂ s cut out by a single formal power series f ζ ∈ O S ̂ s . Similarly, let M ̂ s be the formal completion of M W at 𝑠. Then 𝜌 restricts to a morphism M ̂ s → S ̂ s . Consider the pullback of 𝒜 to 𝖬 and define Def M ⁢ ( ζ , s ) to be the functor defined by (5.3) with S ̂ s replaced by M ̂ s . Then we have a fiber diagram

In particular, Def M ⁢ ( ζ , s ) is a closed formal subscheme of M ̂ s cut out by the pullback ρ ∗ ⁢ ( f ζ ) . Now we prove the key intermediate lemma.

Lemma 5.2.5

Up to replacing 𝜁 by a power, Def M ⁢ ( ζ , s ) is flat over 𝑊.

Proof

It suffices show that if 𝜁 does not lie in the image of L ⁢ ( A M ) Q → L ⁢ ( A s ) Q , then Def M ⁢ ( ζ , s ) is flat over 𝑊, because otherwise, up to replacing 𝜁 by a power, Def M ⁢ ( ζ , s ) = M ̂ s . Suppose by way of contradiction that Def M ⁢ ( ζ , s ) is not flat over 𝑊, which is equivalent to saying that ρ ∗ ⁢ ( f ζ ) vanishes on the entire mod ⁡ p disc M ̂ κ , s : = M ̂ s ⊗ W κ , i.e., the formal completion of M κ at 𝑠. Let 𝜻 be the deformation of 𝜁 over M ̂ κ , s . Let use write 𝑢 for the generic point of M ̂ κ , s . Let 𝑆 be the connected component of M κ which contains 𝑠 and let 𝑣 be its generic point. Since M ̂ κ , s is also the completion of 𝑆 at 𝑠, there is natural embedding k ⁢ ( v ) ↪ k ⁢ ( u ) of residue fields. Let u ̄ be the geometric point over 𝑢 defined by a chosen algebraic closure of k ⁢ ( u ) . We view it also as a geometric point over 𝑣.

Recall that we assumed that Mon ⁢ ( R 2 ⁢ f ét ⁣ ∗ ⁢ Q 2 , b ) is connected in Set-up 5.2.1, so that NS ⁢ ( X η ) Q ↪ NS ⁢ ( X b ) Q is an isomorphism by Lemma 4.1.2. Therefore, π 1 ét ⁢ ( M , b ) acts trivially on NS ⁢ ( X b ) ⊗ Q ℓ 0 . Since α ℓ 0 , b is π 1 ét ⁢ ( M , b ) -equivariant by Definition 5.2.2 (b), π 1 ét ⁢ ( M , b ) acts trivially on L ⁢ ( X b ) ⊗ Q ℓ 0 as well. Hence every element of L ⁢ ( A b ) Q is defined over 𝜂. It follows from [19, I, Proposition 2.7] that L ⁢ ( A η ) = L ⁢ ( A M ) . By Proposition 5.1.4,

dim PNS ⁢ ( X b ) Q = λ geo ⁢ ( P ℓ 0 ) .

Definition 5.2.2 (b) implies that L ℓ 0 | M also has constant λ geo and λ geo ⁢ ( L ℓ 0 | M ) = λ geo ⁢ ( P ℓ 0 ) . Combining these observations, we must have dim L ⁢ ( A M ) Q = λ geo ⁢ ( L ℓ 0 | M ) .

Now consider the subspace

I : = lim → U L ℓ 0 , u ̄ U ,

as 𝑈 runs through open subgroups of π 1 ét ⁢ ( v , u ̄ ) , so that

dim I = λ ⁢ ( L ℓ 0 | S ) = λ geo ⁢ ( L ℓ 0 | M )

(recall Definition 5.1.2). As the endomorphism scheme of an abelian scheme is representable and unramified over the base, every element in L ⁢ ( A u ̄ ) is defined over some finite separable extension of k ⁢ ( v ) inside k ⁢ ( u ̄ ) . Therefore, we obtain a well defined map L ⁢ ( A u ̄ ) ⊗ Q ℓ 0 ↪ I . However, we note that the composite

L ⁢ ( A M ) ⊗ Q ℓ 0 → L ⁢ ( A u ̄ ) ⊗ Q ℓ 0 ↪ I

has to be an isomorphism because dim L ⁢ ( A M ) Q = dim I . This forces the natural map

L ⁢ ( A M ) Q → L ⁢ ( A u ̄ ) Q

to be an isomorphism. Now note that ζ u ̄ ∈ L ⁢ ( A u ̄ ) Q . But as we assumed that 𝜁 does not come from L ⁢ ( A M ) Q , the same has to be true for ζ u ̄ . This gives the desired contradiction. ∎

Suppose we have replaced 𝜁 by a power so that the above lemma holds. Then there exists a DVR 𝑉 finite flat over 𝑊 such that Def M ⁢ ( ζ , s ) admits a 𝑉-valued point (cf. [14, Corollary 1.7]). Let s ̃ ∈ M W ⁢ ( V ) denote the corresponding morphism R → V . Choose an algebraic closure K ̄ . As 𝑉 is a strictly Henselian DVR, it defines an étale path γ s ̃ connecting w : = s ̃ K ̄ and 𝑠. There is a compatible specialization map NS ⁢ ( X w ) → NS ⁢ ( X s ) along 𝑉 (cf. [43, Proposition 3.6] and its proof), which we denote by sp ⁡ ( s ̃ ) . There is a similar specialization map of special endomorphisms. Now we have the following diagram:

Note that char ⁡ K ̄ = 0 , so we have shown that θ w exists. The map θ s does not exist yet. But by construction of s ̃ , 𝜁 lifts to some (necessarily unique) ζ ̃ over 𝑤, and we can define θ s ⁢ ( ζ ) to be sp ⁡ ( s ̃ ) ⁢ ( θ w ⁢ ( ζ ̃ ) ) . Note that θ s ⁢ ( ζ ) does not depend on the choice of s ̃ because its class in P ℓ 0 , s is completely determined by the class of 𝜁 in L ℓ 0 , s via α ℓ 0 , s . Repeating this construction for every ζ ∈ L ⁢ ( A s ) , we obtain the desired map θ s . ∎

5.3 Proof of Theorem B

5.3.1

The reader may have noticed that Theorem 4.1.9 and Theorem 4.1.10 apply to geometrically connected bases. To make use of these results, we consider the following simple-minded functor: if k ⊆ k ′ is a field extension, and 𝑇 is a k ′ -scheme, we write T ( k ) for the 𝑘-scheme obtained by composing the structure morphism of 𝑇 with the projection Spec ⁡ ( k ′ ) → Spec ⁡ ( k ) . Then T ↦ T ( k ) is the left adjoint to the base change functor from 𝑘-schemes to k ′ -schemes.

We will only apply this functor when k = Q . Let 𝑆 be a connected smooth ℚ-variety. Then its scheme of connected components π 0 ⁢ ( S ) is the spectrum of some number field 𝐹. The natural morphism S → π 0 ⁢ ( S ) endows 𝑆 with the structure of an 𝐹-variety. We denote this 𝐹-variety by S F . Note that now S = ( S F ) ( Q ) and S F is geometrically connected as an 𝐹-variety.

5.3.2

Since 𝑆 and S F have the same underlying scheme, étale sheaves and filtered flat vector bundles^[10] on 𝑆 are naturally identified with the corresponding structures on S F and vice versa and we do not distinguish them notationally. If 𝑌 is a ℚ-variety, then there is a canonical bijection between the sets of morphisms ϵ : Mor F ⁢ ( S F , Y F ) ⁢ → ∼ ⁢ Mor Q ⁢ ( S , Y ) . Suppose now that ρ : S F → Y F is a morphism, and 𝑀 is an étale sheaf or a filtered flat vector bundle on 𝑌; then ρ ∗ ⁢ ( M | Y F ) is canonically identified with ϵ ⁢ ( ρ ) ∗ ⁢ M because ϵ ⁢ ( ρ ) = ( Y F → Y ) ∘ ρ as morphisms of schemes.

Proposition 5.3.3

Let U ⊆ Spec ⁢ ( Z ⁢ [ 2 − 1 ] ) be an open subscheme. Let 𝖬 be a connected separated smooth 𝑈-scheme of finite type with generic point 𝜂. Let X → M be a smooth projective morphism such that X | M Q is a ♡ -family.

Let 𝝃 be a relatively ample line bundle of X / M and let Λ ⊆ NS ⁢ ( X η ) Q be a subspace containing the class of ξ η . Let b ∈ M ⁢ ( C ) be a point lying above 𝜂 and let M ∘ be the connected component of M C containing 𝑏. Assume that R 2 ⁢ f ∗ ⁢ Q 2 ⁢ ( 1 ) has constant λ geo , and for every p ∈ U , the pairing on PH 2 ⁢ ( X b , Z ( p ) ) tf is self-dual. Then we have the following.

If ( X / M , ξ ) | M ∘ has maximal monodromy (see Definition 2.2.3), then for every s ∈ M , X s satisfies the Tate conjecture for divisors.
If ( X / M , ξ ) | M ∘ belongs to case (R2^′) (see Section 2.2.4), then up to replacing 𝑈 by a nonempty open subscheme, the above is true.

Here PH 2 ⁢ ( X b , Z ( p ) ) tf denotes the submodule of H 2 ⁢ ( X b , Z ( p ) ) tf consisting of elements orthogonal to the image of Λ in NS ⁢ ( X b ) Q under the pairing on H 2 ⁢ ( X b , Q ) induced by ξ b . Recall that the big monodromy case contains (R+) = (R1) + (R2) and (CM).

Proof

The hypothesis remains unchanged if we replace 𝖬 by a connected étale cover (and 𝑏 by a lift). Therefore, we may assume that Mon ⁢ ( R 2 ⁢ f ∗ ⁢ Q 2 , b ) is connected. Moreover, to prove (b), we may assume that Λ = NS ⁢ ( X b ) Q . Indeed, replacing Λ by NS ⁢ ( X b ) Q might make PH 2 ⁢ ( X b , Z ( p ) ) tf no longer self-dual only for finitely many 𝑝, but we are allowed to shrink 𝑈.

Apply Set-up 4.1.3 to S = M Q and S ∘ = M ∘ . Let V , G = SO ⁢ ( V ) and let μ b : V ⁢ → ∼ ⁢ P B , b be as defined in Section 4.1.6. Let L ⊆ V be the ℤ-lattice defined by μ b − 1 ⁢ ( P B , b ∩ H 2 ⁢ ( X b , Z ) tf ) and choose an N ≫ 0 such that K : = G ( A f ) ∩ ker ( GL ( L ⊗ Z ̂ ) → GL ( L / 2 N L ) ) satisfies condition (♯); in case (b), we additionally require K ℓ 0 = 2 to be sufficiently small (see Section 4.1.6 for these notions). Note that 𝖪 is of the form ∏ q K q , where 𝑞 runs through all primes and K q is a compact open subgroup of G ⁢ ( Q q ) . By Lemma 4.1.4, up to replacing 𝖬 by a further finite connected étale cover, we assume that 𝖪 is admissible, i.e., the image of π 1 ét ⁢ ( M , b ) in GL ⁢ ( P ét , b ) lies in 𝖪 via the chosen isometry μ b . Now we set 𝐹 to be the field such that π 0 ⁢ ( S ) = Spec ⁡ ( F ) and recall that the base point 𝑏 induces an embedding F ↪ C through which S ∘ = ( S F ) C .

For (a), we may now apply Theorem 4.1.9 to S F to conclude that there is a period morphism ρ : S F → Sh K ⁢ ( G ) F together with an isomorphism α B ∘ : ρ C ∗ ⁢ ( V B ) | S ∘ → P B | S ∘ which preserves the Hodge filtrations such that α ét ∘ : = α B ∘ ⊗ A f descends to an isomorphism ρ ∗ ⁢ V ét ⁢ → ∼ ⁢ P ét over S F . By Section 5.3.2, we may view 𝜌 as a morphism S → Sh K ⁢ ( G ) over ℚ and α ét as an isomorphism of étale sheaves over 𝑆. Let p ≠ 2 be a prime in 𝑈. Restrict 𝖬 to Z ( p ) and apply Set-up 5.2.1. Note that, by construction, K p = SO ⁢ ( L ⊗ Z p ) is hyperspecial. Let 𝒮 be the integral model of Sh K ⁢ ( G ) over Z ( p ) . Recall that V ét (resp. P ét ) is the restriction of ∏ ℓ ≠ p L ℓ × L p ⁢ [ 1 / p ] to Sh K ⁢ ( G ) (resp. ∏ ℓ ≠ p P ℓ × P p ⁢ [ 1 / p ] to S = M Q ) (see Section 3.4.2 and Set-up 5.2.1). The existence of α ét allows us to apply Theorem 3.4.4 to extend 𝜌 to a morphism M Z ( p ) → S , which by construction is admissible in the sense of Definition 5.2.2. Now we conclude by Theorem 5.2.3.

For (b), a minor adaptation is needed. Take ℓ 0 = 2 . By Theorem 4.1.10, we still have a period morphism ρ C : S ∘ → Sh K ⁢ ( G ) C equipped with an isomorphism α B ∘ : ρ C ∗ ⁢ V B ⁢ → ∼ ⁢ P B | S ∘ which preserves the Hodge filtrations, but ρ C does not descend all the way to 𝐹. Instead, we only have that, for some finite extension F ′ / F in ℂ, and S ̃ : = ( S F ) ⊗ F F ′ , ρ C descends to a morphism ρ : S ̃ → Sh K ⁢ ( G ) F ′ such that

α ℓ 0 ∘ : = α B ∘ ⊗ Q ℓ 0 descends to an isomorphism ρ ∗ ⁢ V ℓ 0 ⁢ → ∼ ⁢ P ℓ 0 | S ̃ , and
ρ ∗ ⁢ V ℓ is étale locally isomorphic to P ℓ | S ̃ for every other ℓ.

Set S ′ : = S ̃ ( Q ) . Then, by Section 5.3.2 again, we may view 𝜌 as a morphism S ′ → Sh K ⁢ ( G ) and (i), (ii) above as statements about étale sheaves over S ′ . Note that S ′ is naturally a connected étale over of 𝑆. As 𝐿 is self-dual at every p ∈ U , there exists a 𝑈-scheme 𝒮 such that S ⊗ Z ( p ) is the integral canonical model of Sh K ⁢ ( G ) over Z ( p ) (cf. the first theorem of [39]). By a standard spreading-out argument, up to further shrinking 𝑈, we may assume that S ′ is the ℚ-fiber of a smooth 𝑈-scheme M ′ ; moreover, S ′ → S extends to an étale covering map M ′ → M and ρ : S ′ → Sh K ⁢ ( G ) extends to a morphism ρ U : M ′ → S . The reader readily checks using (i) and (ii) above that, for each p ∈ U , the localization ρ U ⊗ Z ( p ) is an ℓ 0 -admissible period morphism in the sense of Definition 5.2.2. Therefore, the conclusion follows from Theorem 5.2.3. ∎

We are now ready to prove Theorem B.

Proof

Note that the conclusion of Theorem B is for p ≫ 0 . By Proposition 5.3.3, it suffices to show that there are exists an open subscheme U ⊆ Spec ⁡ ( Z ⁢ [ 2 − 1 ] ) such that, after we restrict 𝖬 to 𝑈, R 2 ⁢ f ∗ ⁢ Q 2 has constant λ geo . By combining Nagata’s compactification and Hironaka’s resolution of singularities in characteristic zero, we can find a compactification M ̄ of the generic fiber M Q such that the boundary D : = M ̄ − M Q , equipped with the reduced scheme structure, is a normal crossing divisor. For some open subscheme 𝑈 of Spec ⁢ ( Z ⁢ [ 2 − 1 ] ) , M ̄ and 𝔇 are defined over 𝑈, and 𝔇 becomes a relative normal crossing divisor over 𝑈. This implies that, for ( p ) ∈ U , M Z ( p ) admits a good compactification relative to Z ( p ) in the sense of Definition 5.1.2. Now we conclude by Lemma 5.1.3. ∎

To effectively apply Proposition 5.3.3, the crux of the matter is to verify the hypothesis that R 2 ⁢ f ∗ ⁢ Q 2 ⁢ ( 1 ) has constant λ geo . In practice, we achieve this by finding suitable proper curves on some partial compactification of 𝖬. This is encapsulated in the following lemma, which is not stated in full generality but is tailored for direct application to the concrete situations we will consider.

Lemma 5.3.4

Let 𝑝 be a prime and ℓ a different prime. Set W : = W ( F ̄ p ) , K : = W [ 1 / p ] and choose an isomorphism K ̄ ≅ C . Let 𝖬 be a connected separated smooth Z ( p ) -scheme of finite type such that M F p and M Q are both geometrically irreducible. Let X → M be a smooth projective morphism such that X | M Q is a ♡ -family. Let 𝖵 be the VHS defined on R 2 ⁢ f C ⁣ ∗ ⁢ Q ⁢ ( 1 ) .

Suppose that there exists a smooth connected 𝑊-curve 𝐶 with a morphism to 𝖬 such that

the image of C C is not contained in the Noether–Lefschetz loci of 𝖵 and the restriction V | C C is non-isotrivial, and
𝐶 has a good compactification over 𝑊.

Then L : = R 2 f ∗ Q ℓ ( 1 ) has constant λ geo .

Proof

Note that condition (i) implies λ ⁢ ( L | C C ) = λ ⁢ ( L | M C ) , due to Proposition 5.1.4. Choose a base point s ∈ M ⁢ ( C ) which lies over the generic point 𝜂 of 𝖬. Let s p be a geometric point over the generic point η p of M F ̄ p . Since 𝑠 specializes to s p ,

dim NS ⁢ ( X s p ) Q ≥ dim NS ⁢ ( X s ) Q ;

moreover, as every element in NS ⁢ ( X s p ) Q is stabilized by an open subgroup of π 1 ét ⁢ ( M F ̄ p , s p ) ,

λ ⁢ ( L | M F ̄ p ) ≥ dim NS ⁢ ( X s p ) Q .

On the other hand, as π 1 is a functor, we have λ ⁢ ( L | M F ̄ p ) ≤ λ ⁢ ( L | C F ̄ p ) by default. But the curve L | C has constant λ geo because 𝐶 has a good relative compactification (Lemma 5.1.3), so we have

dim NS ⁢ ( X s ) Q ⁢ = Proposition 5.1.4 ⁢ λ ⁢ ( L | M C ) = λ ⁢ ( L | C C ) = λ ⁢ ( L | C F ̄ p ) ≥ λ ⁢ ( L | M F ̄ p ) .

This implies that λ ⁢ ( L | M C ) = λ ⁢ ( L | M F ̄ p ) , as desired. ∎

6 Deforming curves on parameter spaces

In this section, we prepare some preliminary lemmas which will be used to construct the curves as in Lemma 5.3.4 on appropriate moduli spaces.

6.1 Families of curves which homogeneously dominate a variety

Let 𝑘 be an algebraically closed field. For a morphism f : X → Y between 𝑘-varieties, we say that 𝑓 has equi-dimensional fibers if, for every two points y , y ′ ∈ Y , dim X y = dim X y ′ . If f : X → Y is a smooth morphism between 𝑘-varieties, denote by T ⁢ ( X / Y ) the relative tangent bundle, i.e., the dual of Ω X / Y 1 . If Y = Spec ⁡ ( k ) , then we simply write T ⁢ X for T ⁢ ( X / Y ) , and for a 𝑘-point x ∈ X , we write T x ⁢ X for the tangent space T x ⁢ X to emphasize that it is a fiber of T ⁢ X .

Definition 6.1.1

Let 𝑆 and 𝑇 be two smooth irreducible 𝑘-varieties, let g : C → T be a smooth family of connected curves, and let φ : C → S be a morphism with equi-dimensional fibers (i.e., there exists some integer d ≥ 0 such that every geometric fiber of 𝜑 is nonempty and equi-dimensional of dimension 𝑑). Let 𝑈 be the maximal open subvariety of 𝒞 on which the composition T ⁢ ( C / T ) ↪ T ⁢ C → φ ∗ ⁢ ( T ⁢ S ) does not vanish. Suppose that

the induced morphism C → T × S is quasi-finite,
for every 𝑘-point s ∈ S , U s : = U ∩ φ − 1 ( s ) is dense in φ − 1 ⁢ ( s ) ;
the morphism U → P ⁢ ( T ⁢ S ) has equi-dimensional fibers.

Then we say that the family of curves C / T homogeneously dominates𝑆 (via the morphism 𝜑). If there exists an open dense subvariety T ′ ⊆ T such that the restriction C | T ′ homogeneously dominates 𝑆, then we say that C / T strongly dominates 𝑆.

The natural morphism U → P ⁢ ( T ⁢ S ) is induced by the identification C ≅ P ⁢ ( T ⁢ ( C / T ) ) . Roughly speaking, the family C / T homogeneously dominates 𝑆 if there are curves parameterized by 𝑇 passing through every given point on 𝑆 in any given direction, and the subfamily of such curves has a fixed dimension.

The notion “ C / T strongly dominates 𝑆” is only defined for convenience, as sometimes the natural families of curves have some bad locus on 𝑇 of smaller dimension which does not affect applications.

Lemma 6.1.2

Let P → S be a smooth morphism between smooth 𝑘-varieties. Let X ⊆ P be a relative effective Cartier divisor whose total space is smooth. If s ∈ S ⁢ ( k ) is a point such that X s has isolated singularities, then there exists an open dense subvariety U ⊆ P ⁢ ( T s ⁢ S ) with the following property: for every unramified morphism φ : C → S from a smooth curve 𝐶 which sends a point c ∈ C to 𝑠, the total space of the pullback family X | C has no singularity on X s if d ⁢ φ ⁢ ( P ⁢ ( T c ⁢ C ) ) ∈ U .

Proof

Since the question is étale-local in nature, we might as well assume that S = A k m for m = dim S , s = 0 , X s has a single isolated singularity at a 𝑘-point P ∈ X s ⊆ P s , and 𝒫 is isomorphic to A S n = S × A k n near 𝑃. Let x i ’s and s j ’s be the coordinates on A k n and S k respectively. Suppose that 𝒳 is locally cut out by an equation F ⁢ ( x 1 , … , x n , s 1 , … , s m ) near 𝑃. The assumption that 𝑃 is a singularity of the fiber X s but not of the total space 𝒳 implies that ∂ F / ∂ s j ≠ 0 at 𝑃 for some 𝑗. One may simply take 𝑈 to be the open subscheme of P ⁢ ( T s ⁢ S ) ≅ P m − 1 where the coordinate of ∂ s j is nonzero. ∎

Definition 6.1.3

Let f : X → S be a finite type morphism between schemes.

Let the singular locus Sing ⁢ ( f ) be the reduced closed subscheme of 𝑋 whose support consists of all points where 𝑓 fails to be smooth.^[11]
If 𝑓 is in addition proper and flat, we say that the scheme-theoretic image of Sing ⁢ ( f ) is the (generalized) discriminant locus of 𝑓, and denote it by Disc ⁢ ( f ) .
In the above situation, we say that Disc ⁢ ( f ) is mild if it has codimension at least 1 in 𝑆 and there exists a dense open subscheme V ⊆ Disc ⁢ ( f ) such that, for every geometric point 𝑠 on 𝑉, the fiber X s has only isolated singularities.

Remark 6.1.4

Note that the properness assumption on 𝑓 implies that Disc ⁢ ( f ) is closed in 𝑆. Moreover, since it is defined to be the scheme-theoretic image of a reduced scheme, it is also reduced. Its formation commutes with flat base change but not arbitrary base change: for any morphism T → S , Disc ⁢ ( f T ) is always the reduced subscheme of Disc ⁢ ( f ) T , so they are equal if and only if the latter is reduced.

Remark 6.1.5

In many natural settings, the discriminant locus Disc ⁢ ( f ) is expected to have codimension 1. However, no general theorem currently characterizes precisely when this is true. We provide an ad hoc criterion which suffices for our purposes. Suppose that the base 𝑆 is an open subvariety of P C n for some n ≥ 1 and the complement has codimension at least 2. Let f : X → S be a proper and generically smooth morphism. If, over U : = S \ Disc ( f ) , the VHS on R i ⁢ ( f U ) ∗ ⁢ Q for some i ∈ N is non-isotrivial, then Disc ⁢ ( f ) must have codimension 1. Indeed, otherwise, 𝑈 is simply connected and cannot carry a non-isotrivial VHS.

Proposition 6.1.6

Let 𝒳 and 𝑆 be as in Lemma 6.1.2. Suppose that

the generalized discriminant variety D : = Disc ( f ) ⊆ S is mild;
there is a family of smooth curves g : C → T which homogeneously dominates 𝑆 through a morphism φ : C → S .

Then, for a general 𝑘-point t ∈ T k , the total space of the pullback family X | C t is smooth.

Proof

By assumption, we have a diagram

Let Z ⊆ S × T be the subset of points ( s , t ) such that C t passes through 𝑠 and the total space of X | C t has a singularity lying above φ t − 1 ⁢ ( s ) . It is easy to see that 𝒵 is constructible. Let X | C be the pullback of 𝒳 along 𝜑. Then we have a natural morphism X | C → T × k S and 𝒵 is the set-theoretic image of Sing ⁢ ( X | C → T ) . Endow 𝒵 with the structure of a reduced scheme.

It suffices to show dim Z < dim T . Let 𝑉 be as in Definition 6.1.3 (c) and for s ∈ V ⁢ ( k ) , let 𝑈 and U s = U ∩ φ − 1 ⁢ ( s ) be as introduced in Definition 6.1.1). Let t = g ⁢ ( s ) . By Lemma 6.1.2, there exists a proper closed subvariety U s , bad ⊆ U such that the total space of X | C t is not smooth near the fiber X u only if u ∈ U s , bad . Let Z ̃ s ⊆ φ − 1 ⁢ ( s ) be the union of the complement of U s and the Zariski closure of U s , bad . Then Z ̃ s is a proper closed subvariety of φ − 1 ⁢ ( s ) . Since the morphism φ − 1 ⁢ ( s ) → T is quasi-finite and the fiber Z s ⊆ T is contained in the image of Z ̃ s , we have dim Z s < dim φ − 1 ⁢ ( s ) = dim C − dim S . Since the image of 𝒵 in 𝑆 is contained in 𝐷, and 𝑠 runs over an open dense subvariety of 𝐷, we have that dim Z ≤ dim C − 2 = dim T − 1 , as desired. ∎

6.2 Applications of the Baire category theorem

Let 𝑘 be an algebraically closed field of characteristic p > 0 . Set W : = W ( k ) and K : = W [ 1 / p ] . Choose an algebraic closure K ̄ of 𝐾.

Lemma 6.2.1

Suppose that S → B is a flat morphism between irreducible smooth 𝑊-schemes of finite type. Let 𝑁 be a countable union of closed proper subschemes of S K ̄ . Let b ∈ B ⁢ ( k ) be any point and let B ̂ b be the formal completion of 𝐵 at 𝑏. Then the subset of points b ̃ ∈ B ̂ b ⁢ ( W ) such that supp ⁢ ( S b ̃ ⊗ K ̄ ) is not contained in 𝑁 is analytically dense.

Proof

Let U : = B ̂ b ( W ) . By taking the union of 𝑁 with all its Galois conjugates, we may assume that 𝑁 is defined over 𝐾. Let N 1 , N 2 , … be the irreducible components of 𝑁. By flatness, the morphism S → B is open, so for each 𝑖, there exists a proper closed subscheme Z i ⊆ B such that every z ∈ B ⁢ ( K ) satisfies supp ⁡ ( S z ) ⊆ N i only if z ∈ Z i . Indeed, one may simply take Z i to be the complement of the image of S − N i . Since each U − Z i ⁢ ( W ) is open dense in analytic topology, we may conclude by the Baire category theorem for complete metric spaces that U − ⋃ i = 1 ∞ Z i ⁢ ( W ) is analytically dense. ∎

Lemma 6.2.2

Let 𝑆 and 𝑇 be smooth irreducible 𝑊-schemes of finite type and let 𝑁 be a countable union of closed proper subschemes of S K ̄ . Let t ∈ T k be a closed point and T ̂ t the formal completion of 𝑇 at 𝑡.

Suppose that 𝒞 is a smooth family of geometrically connected curves over 𝑇, and there is a morphism φ : C → S such that the family ( C / T ) K ̄ over K ̄ strongly dominates S K ̄ . Then the subset of points t ̃ ∈ T ̂ t ⁢ ( W ) such that φ ⁢ ( supp ⁡ ( C t ̃ ⊗ K ̄ ) ) is not contained in 𝑁 is analytically dense.

Proof

Again by Galois descent, we may assume that N = ⋃ i = 1 ∞ N i for irreducible closed subschemes N i of S K . Let M i : = C K × φ N i and M : = ⋃ i = 1 ∞ M i . The assumption that ( C / T ) K ̄ strongly dominates S K ̄ implies that each M i is a proper closed subscheme. Now we apply the above lemma with ( S → B , N ) replaced by ( C → T , M ) . ∎

7 Elliptic surfaces with p g = q = 1

7.1 Generalities on elliptic surfaces

In this section, we recall some basic facts about elliptic surfaces and describe their moduli. The goal is to establish Lemma 7.3.3, which will be used in conjunction with the lemmas in the preceding section, so that eventually we can apply Theorem 5.2.3 and Lemma 5.3.4 together.

Let 𝑘 be an algebraically closed field of characteristic ≠ 2 , 3 . Let 𝐶 be a smooth projective curve over 𝑘 and let π : X → C be an elliptic surface over 𝐶 with a zero section σ : C → X through which we also view 𝐶 as a curve on 𝑋. The fundamental line bundle𝐿 of X / C is defined to be the dual of the normal bundle N C / X , or equivalently that of R 1 ⁢ π ∗ ⁢ O X . The degree of 𝐿 is defined to be the height of X / C (or rather its generic fiber), which we denote by h ⁢ ( X ) . Set V r = H 0 ⁢ ( L r ) . There exists a pair ( a 4 , a 6 ) ∈ V 4 × V 6 − { 0 } , which is unique up to the action of λ ∈ k × by λ ⋅ ( a 4 , a 6 ) = ( λ 4 ⁢ a 4 , λ 6 ⁢ a 6 ) , such that 𝑋 is the minimal resolution of the hypersurface X ′ ⊆ P ⁢ ( L 2 ⊕ L 3 ⊕ O C ) defined by the Weierstrass equation (see [30, Theorem 1])

(7.1) y 2 ⁢ z = x 3 − a 4 ⁢ x ⁢ z 2 − a 6 ⁢ z 3 ,

where x , y , z are homogeneous coordinates on L 2 , L 3 , O C respectively. The hypersurface X ′ has at most rational double point singularities and is called the Weierstrass normal form of the original surface 𝑋. If X ′ is smooth, then of course, X = X ′ . In this paper, we only consider 𝑋 with h ⁢ ( X ) > 0 .

Next, we recall that Kodaira classified all the possible singular fibers in the elliptic fibration π : X → C when k = C in [36], and his classification is well known to hold verbatim in characteristic ≠ 2 , 3 as well. We refer the reader to [56, §4] for a summary. Set Δ : = 4 a 4 3 − 27 a 6 2 . Let c ∈ C be a point and denote by val c the valuation defined by a uniformizer at 𝑐. The only facts we shall need from [56, §4] are the following.

Proposition 7.1.1

The following statements hold.

The fiber X c is singular if and only if Δ vanishes at 𝑐, i.e., val c ⁢ ( Δ ) ≥ 1 .
X c is of I n type ( n > 0 ) if and only if val c ⁢ ( a 4 ) = val c ⁢ ( a 6 ) = 0 , and n = val c ⁢ ( Δ ) .
X c is of II -type if and only if val c ⁢ ( a 4 ) ≥ 1 and val c ⁢ ( a 6 ) = 1 .
If X c is a singular fiber of any other type, val c ⁢ ( Δ ) ≥ 3 .
If X c is of I 1 -type or II -type, then X c = X c ′ . In other words, the singularity on X c ′ is not a surface singularity.
If X c is of I 2 -type, then X c ′ has a unique ODP singularity given by contracting the irreducible component not meeting the zero section.

The degree of the discriminant Δ is 12 ⁢ χ ⁢ ( O X ) = e ⁢ ( X ) . Recall that the genus g ⁢ ( C ) is equal to the irregularity q ⁢ ( X ) and we have p g ⁢ ( X ) = χ ⁢ ( O X ) − 1 + g ⁢ ( C ) . Therefore, elliptic surfaces with p g = 1 fall into two types.

χ ⁢ ( O X ) = 2 and g ⁢ ( C ) = 0 . These are elliptic K3 surfaces.
χ ⁢ ( O X ) = g ⁢ ( C ) = 1 . These surfaces have Kodaira dimension 1.

We are interested in the latter class. Note that, although these surfaces are elliptic fibrations over genus 1 curves, one should not confuse them with bielliptic surfaces, which are of Kodaira dimension 0.

7.1.2

For future reference, we introduce some notation. Let 𝐵 be a base scheme and 𝒱 a vector bundle over 𝐵. We denote by A ⁢ ( V ) the relative affine space over 𝐵 defined by 𝒱 and by A ⁢ ( V ) ∗ the open part of A ⁢ ( V ) minus the zero section. Given a sequence of numbers q = ( q 0 , … , q m ) such that q i ’s are invertible in O B and vector bundles V 0 , … , V m such that V = ⨁ i = 0 m V i , we denote by P q ⁢ ( V ) the resulting weighted projective stack, i.e., the quotient stack of G m -action on 𝒱 given by

λ : ( v 0 , … , v m ) ↦ ( λ q 0 ⁢ v 0 , … , λ q m ⁢ v m ) ⁢ for ⁢ λ ∈ G m ,

and by P q ⁢ ( V ) the coarse moduli space of P q ⁢ ( V ) . It is well known that this coarse moduli space can be constructed explicitly by applying the relative Proj functor to a sheaf of graded algebras over 𝐵. We omit the details. If 𝑞 is not specified, then it is assumed to be ( 1 , … , 1 ) .

Set-up 7.1.3

Let 𝐵 be a Noetherian Z ⁢ [ 1 / 6 ] -scheme, let ϖ : C → B be a family of smooth projective curves over 𝐵 of genus 𝑔, and let ℒ be a relative line bundle on 𝒞 of degree ℎ. Assume that 4 ⁢ h ≥ 2 ⁢ g − 1 and h ≥ 1 . Let V r denote the vector bundle ϖ ∗ ⁢ L r for r ≥ 4 . Let X ̃ be the subscheme of

A ⁢ ( V 4 ⊕ V 6 ) ∗ × B P ⁢ ( L 2 ⊕ L 3 ⊕ O C )

defined by the Weierstrass equation (7.1) in the obvious way. Let 𝜇 be the G m -action on A ⁢ ( V 4 ⊕ V 6 ) ∗ × B P ⁢ ( L 2 ⊕ L 3 ⊕ O C ) defined by

λ ⋅ ( ( a 4 , a 6 ) , [ x : y : z ] ) = ( ( λ 4 a 4 , λ 6 a 6 ) , [ λ 2 x : λ 3 y : z ] ) for λ ∈ G m .

Let Q ⁢ ( μ ) denote the quotient stack of the G m -action 𝜇. Then X ̃ descends to an algebraic substack 𝒳 of Q ⁢ ( μ ) . Note that Q ⁢ ( μ ) , and hence 𝒳, admit natural morphisms to P ( 4 , 6 ) ⁢ ( V 4 ⊕ V 6 ) . Set D : = Disc ( X ̃ / A ( V 4 ⊕ V 6 ) ∗ ) (see Definition 6.1.3). It defines a closed substack D ̄ in P ( 4 , 6 ) ⁢ ( V 4 ⊕ V 6 ) , because it is invariant under the G m -action on A ⁢ ( V 4 ⊕ V 6 ) ∗ through weight ( 4 , 6 ) . Let 𝔘 denote the open complement of 𝔇 in A ⁢ ( V 4 ⊕ V 6 ) ∗ .

Remark 7.1.4

Note that, with fixed ( C , L ) , the formation of V 4 , V 6 , X ̃ , 𝒳, and 𝔘 naturally commutes with base change among 𝐵-schemes. We will implicitly use this for the rest of Section 7. However, a priori, 𝔇 and D ̄ might not commute with non-flat base change as they can become non-reduced (cf. Remark 6.1.4). Much of Section 7 is devoted to giving conditions to exclude this possibility. The key intermediate result is Lemma 7.3.3, which will play an important role in the proof of Theorem A.

We remark that 𝒳 is only “stacky” because of the base.

Lemma 7.1.5

Let 𝑇 be a Noetherian 𝐵-scheme and let ν : T → P ( 4 , 6 ) ⁢ ( V 4 ⊕ V 6 ) be a morphism. Then the pullback ν ∗ ⁢ Q ⁢ ( μ ) , and hence ν ∗ ⁢ X , are flat projective schemes over 𝑇.

Proof

The reader can check that Q ⁢ ( μ ) is in fact a P 2 -bundle over

P ( 4 , 6 ) ⁢ ( V 4 ⊕ V 6 ) × B C .

Therefore, the pullback ν ∗ ⁢ Q ⁢ ( μ ) is a P 2 -bundle over the scheme T × B C . Being a closed substack of the scheme ν ∗ ⁢ Q ⁢ ( μ ) , ν ∗ ⁢ X has to be a projective scheme. The flatness of ν ∗ ⁢ Q ⁢ ( μ ) is clear, and one deduces the flatness of ν ∗ ⁢ X using that it is locally cut out by a single equation, and its geometric fibers are of codimension 1 (cf. [66, Tag 00MF]). ∎

Proposition 7.1.6

The morphism X ̃ → B is smooth.

Proof

As X ̃ is flat over 𝐵, it suffices to check smoothness of geometric fibers. Hence we may assume that B = Spec ⁢ ( k ) , where 𝑘 is an algebraically closed field of characteristic ≠ 2 , 3 . Let us simply write A ∗ for A ⁢ ( V 4 ⊕ V 6 ) ∗ . Choose a point u ∈ A ∗ ⁢ ( k ) and c ∈ C ⁢ ( k ) . Choose a uniformizer 𝑡 of 𝐶 at 𝑐 and bases { σ i } , { θ j } for V 4 , V 6 respectively. Then the formal completion of A ∗ × C at ( u , c ) can be identified with Spf ⁢ ( R ) , where

R = k ⁢ ⟦ t , α 0 , … , α 4 ⁢ h − g , β 0 , … , β 6 ⁢ h − g ⟧ .

By choosing a local O C -generator of 𝐿 at 𝑐, we turn σ i ’s and θ j ’s into elements in k ⟦ t ⟧ . Let ( { a i } , { b j } ) ∈ k 10 ⁢ h − 2 ⁢ g + 2 be the affine coordinates of 𝑢 in A ⁢ ( V 4 ⊕ V 6 ) . Then the restriction of X ̃ to Spf ⁢ ( R ) can be identified with the subscheme of Proj ⁡ R ⁢ [ x , y , z ] defined by the equation

W : = y 2 z − x 3 + ( ∑ i = 0 4 ⁢ h − g ( a i + α i ) σ i ( t ) ) x z 2 + ( ∑ j = 0 6 ⁢ h − g ( b j + β j ) θ j ( t ) ) z 3 = 0 .

Let 𝑟 be the special point of Spf ⁢ ( R ) . The singularity of the (generalized) elliptic curve X ̃ t defined by the above equation when 𝑡, α i , and β j all vanish cannot appear on the z = 0 chart. So we may set z = 1 in the above equation and consider the resulting scheme in Spec ⁢ R ⁢ [ x , y ] . As θ j ’s form a basis of V 6 , θ j ⁢ ( 0 ) ≠ 0 for some 𝑗. Then, for this 𝑗, the partial derivative ∂ W / ∂ β j remains nonzero on the special fiber. This implies that the total space of the restriction of X ̃ to Spf ⁢ ( R ) is smooth. But the choice of ( u , c ) is arbitrary, so X ̃ is smooth. ∎

7.2 Nonlinear Bertini theorems for families of elliptic surfaces

In this section, 𝑘 remains an algebraically closed field of characteristic ≠ 2 , 3 .

Proposition 7.2.1

Let 𝐶 be a smooth projective curve of genus 𝑔 over 𝑘 and let 𝐿 be a line bundle on 𝐶 with degree ℎ. Set V r : = H 0 ( L r ) for every r ∈ N . For every d ∈ N , consider the closed subset of A ⁢ ( V 4 ⊕ V 6 ) × C defined by ( Δ ∈ V 12 is defined by 4 ⁢ a 4 3 − 27 ⁢ a 6 2 as before)

K d : = { ( a 4 , a 6 , c ) ∈ V 4 × V 6 × C ∣ val c ( Δ ) ≥ d }

and endow it with the reduced subscheme structure. Likewise, let D ⊆ C × C be the diagonal and define a closed subscheme in A ⁢ ( V 4 ⊕ V 6 ) × ( C × C − D ) by

K 2 + : = { ( a 4 , a 6 , c , c ′ ) ∈ V 4 × V 6 × ( C × C − D ) ∣ val c ( Δ ) and val c ′ ( Δ ) are both ≥ 2 } .

If 2 ⁢ h ≥ g + 1 , then we have the following:

K d has codimension 𝑑 for d ≤ 3 .
K 2 has two irreducible components K 2 ⁢ ( I 2 ) and K 2 ⁢ ( II ) characterized by conditions val c ⁢ ( a 6 ) = 0 and val c ⁢ ( a 6 ) ≥ 1 respectively.
K 2 + has codimension 4.

Proof

Recall that, by the Riemann–Roch theorem, for any line bundle 𝑀 on 𝐶,

if deg ⁡ ( M ) ≥ 2 ⁢ g − 1 , then h 0 ⁢ ( M ) = deg ⁡ ( M ) − g + 1 ;
if deg ⁡ ( M ) = 2 ⁢ g − 2 , then h 0 ⁢ ( M ) = deg ⁡ ( M ) − g + 1 unless M ≅ ω C .

Fix any point c ∈ C and consider the projection K d → C . It suffices to show that the fiber K d , c over 𝑐, viewed naturally as a closed subscheme of A ⁢ ( V 4 ⊕ V 6 ) , has codimension 𝑑. We identify the completion of 𝐶 along 𝑐 with Spf ( k ⟦ t ⟧ ) by choosing a uniformizer 𝑡. After choosing a local generator of 𝐿, we may consider the Taylor series of any σ ∈ H 0 ⁢ ( L r ) , which is a power series σ ( t ) ∈ k ⟦ t ⟧ . By the first paragraph, for r ≥ 4 and d ≤ 3 , we have

h 0 ⁢ ( L r ⁢ ( ( 1 − d ) ⁢ c ) ) = h 0 ⁢ ( L r ⁢ ( − d ⁢ c ) ) + 1 .

Therefore, we may choose a basis σ 0 , … , σ 4 ⁢ h − g for V 4 such that val c ⁢ ( σ i ) = i for i = 0 , 1 , 2 and { σ i } 3 ≤ i ≤ 4 ⁢ h − g forms a basis for H 0 ⁢ ( L 4 ⁢ ( − 3 ⁢ c ) ) . We may assume that σ 0 ⁢ ( t ) ≡ 1 , σ 1 ⁢ ( t ) ≡ t and σ 2 ⁢ ( t ) ≡ t 2 modulo t 3 . We choose a basis { θ 0 , … , θ 6 ⁢ h − g } in an entirely similar way.

With the given choices of bases, we use { α i } and { β j } for the coordinates of V 4 and V 6 respectively, so that Δ can be expressed as

(7.2) Δ = 4 ⁢ ( ∑ i = 0 4 ⁢ h − g α i ⁢ σ i ) 3 − 27 ⁢ ( ∑ j = 0 6 ⁢ h − g β j ⁢ θ j ) 2 .

Then the fiber K d , c ( d ≤ 3 ) is supported on the subset of A ⁢ ( V 4 ⊕ V 6 ) cut out by the first 𝑑 equations from below:

{ Δ ⁢ ( 0 ) = 4 ⁢ α 0 3 − 27 ⁢ β 0 2 , Δ ′ ⁢ ( 0 ) = 3 ⁢ ( 4 ⁢ α 0 2 ⁢ α 1 − 18 ⁢ β 0 ⁢ β 1 ) , Δ ′′ ⁢ ( 0 ) = 24 ⁢ ( α 0 ⁢ α 1 2 + α 0 2 ⁢ α 2 ) − 54 ⁢ ( β 1 2 + 2 ⁢ β 0 ⁢ β 2 ) .

Statement (a) is clear for d = 0 , 1 . For d = 2 , it is clear that K 2 contains the following subscheme:

K 2 ( II ) : = { ( a 4 , a 6 , c ) ∈ V 4 × V 6 × C ∣ val c ( a 4 ) ≥ 1 , val c ( a 6 ) ≥ 1 } ,

such that the support of the fiber of K 2 ⁢ ( II ) over 𝑐 is cut out by α 0 = β 0 = 0 . Let

A 2 : = A ( k ⟨ σ 0 , θ 0 ⟩ )

be the affine space with coordinates ( α 0 , β 0 ) and let C ′ ⊆ A 2 be the cuspidal curve defined by Δ ⁢ ( 0 ) = 0 . Then the fiber of K 2 − K 2 ⁢ ( II ) over a point in C ′ − { ( 0 , 0 ) } is given by a codimension 1 hyperplane in A ⁢ ( k ⁢ ⟨ σ i , β j ⟩ i , j ≥ 1 ) . This implies that K 2 − K 2 ⁢ ( II ) is irreducible of codimension 2 in A ⁢ ( V 4 ⊕ V 6 ) , and we denote this component by K 2 ⁢ ( I 2 ) . Note that this implies (b). To see the d = 3 case for (a), just note that Δ ′′ ⁢ ( 0 ) does not vanish identically on both K 2 ⁢ ( I 2 ) and K 2 ⁢ ( II ) .

Finally, we treat (c). We consider the projection Φ : K 2 + → ( C × C − D ) and take a point ( c , c ′ ) ∈ ( C × C − D ) . Denote the fiber of Φ over ( c , c ′ ) by Φ ( c , c ′ ) . We assume first that L 4 ≇ ω C ⁢ ( 2 ⁢ c + 2 ⁢ c ′ ) . This condition is automatically satisfied when 2 ⁢ h > g + 1 and ensures that h 0 ⁢ ( L r ⁢ ( − 2 ⁢ c − 2 ⁢ c ′ ) ) = r ⁢ h + g − 5 for r ≥ 4 . Then we may choose σ 0 , … , σ 3 ∈ V 4 with the following vanishing orders:

	σ 0	σ 1	σ 2	σ 3
val c	0	1	≥ 2	≥ 2
val c ′	≥ 2	≥ 2	0	1

We complete { σ 0 , … , σ 3 } to a basis { σ i } of V 4 by adjoining a basis for H 0 ⁢ ( L 4 ⁢ ( − 2 ⁢ c − 2 ⁢ c ′ ) ) . Let t , s be uniformizers of the completions of 𝐶 along 𝑐 and c ′ respectively. After choosing local generators of 𝐿, we may consider Taylor series σ i ( t ) ∈ k ⟦ t ⟧ and σ i ( s ) ∈ k ⟦ s ⟧ , and assume that σ 0 ⁢ ( t ) ≡ 1 , σ 1 ⁢ ( t ) ≡ t mod t 2 and σ 2 ⁢ ( s ) ≡ 1 , σ 3 ⁢ ( s ) ≡ s mod s 2 . Choose an entirely similar basis { θ j } for V 6 and express Δ again as in (7.2). Then the defining equations for Φ ( c , c ′ ) in A ⁢ ( V 4 ⊕ V 6 ) are

{ 4 ⁢ α 0 3 − 27 ⁢ β 0 2 = 4 ⁢ α 2 3 − 27 ⁢ β 2 2 = 0 , 3 ⁢ ( 4 ⁢ α 0 2 ⁢ α 1 − 18 ⁢ β 0 ⁢ β 1 ) = 3 ⁢ ( 4 ⁢ α 2 2 ⁢ α 3 − 18 ⁢ β 2 ⁢ β 3 ) = 0 .

By the same argument for the d = 2 case in (a), the above equations define a codimension 4 subscheme. The point is that the variables with indices 0 , 1 do not interfere with those with 2 , 3 .

It remains to deal with the case when 2 ⁢ h = g + 1 and L 4 ≅ ω C ⁢ ( 2 ⁢ c + 2 ⁢ c ′ ) . Note that, in this case, g ≥ 1 , so the condition

L 4 ≅ ω C ⁢ ( 2 ⁢ c + 2 ⁢ c ′ )

defines a closed subscheme of ( C × C − D ) of codimension at least 1. Therefore, it is enough to show that the codimension of Φ ( c , c ′ ) is at least 3. Note that we are able to choose a basis { θ j } just as before, but this time choose { σ 0 , σ 1 , σ 2 } with the following vanishing orders:

	σ 0	σ 1	σ 2
val c	0	1	2
val c ′	2	1	0

and complete it to a basis of V 4 by adjoining a basis of H 0 ⁢ ( L 4 ⁢ ( − 2 ⁢ c − 2 ⁢ c ′ ) ) . Assume that σ 0 ⁢ ( t ) = 1 , σ 1 ⁢ ( t ) = t mod t 3 , and σ 2 ⁢ ( s ) = 1 mod s . Then the conditions

val c ⁢ ( Δ ) ≥ 2 and val c ′ ⁢ ( Δ ) ≥ 1

give us 3 equations which are necessarily satisfied by Φ ( c , c ′ ) :

{ 4 ⁢ α 0 3 − 27 ⁢ β 0 2 = 4 ⁢ α 2 3 − 27 ⁢ β 2 2 = 0 , 3 ⁢ ( 4 ⁢ α 0 2 ⁢ α 1 − 18 ⁢ β 0 ⁢ β 1 ) = 0 .

It is clear that these indeed cut out a subscheme of codimension 3. ∎

Proposition 7.2.2

Assume 2 ⁢ h ≥ g + 1 . Apply Set-up 7.1.3 to B : = Spec ( k ) . The resulting discriminant 𝔇 is a proper subvariety of A ⁢ ( V 4 ⊕ V 6 ) ∗ . If codim ⁡ D = 1 , then 𝔇 has a unique irreducible component D 0 of maximal dimension; moreover, for a general point 𝑎 on D 0 , X ̃ a is smooth away from a single ODP.

Proof

Because B = Spec ⁡ ( k ) , ( V 4 , V 6 , C ) above is the same as ( V 4 , V 6 , C ) in Proposition 7.2.1, and we use the notation from Proposition 7.2.1 and the results in Proposition 7.1.1 throughout the proof below.

Note that, for a = ( a 4 , a 6 ) ∈ A ⁢ ( V 4 ⊕ V 6 ) ∗ such that Δ a = 4 ⁢ a 4 3 − 27 ⁢ a 6 3 ∈ H 0 ⁢ ( L 12 ) does not vanish identically on 𝐶, X ̃ a is singular if and only if its elliptic fibration has a reducible singular fiber. It is clear that 𝔇 is contained in the image of K 2 in A ⁢ ( V 4 ⊕ V 6 ) , and hence has codimension at least 1. If codim ⁡ D = 1 , then by Proposition 7.2.1 (c), there exists an open dense subset U ⊆ D such that if a ∈ U , X ̃ a has at most one singular fiber not of I 1 -type. If moreover this singular fiber is of II -type, then X ̃ a is smooth and a ∉ D . Therefore, the only possible irreducible component of maximal dimension in 𝔇 is the Zariski closure of the image of K 2 ⁢ ( I 2 ) . This implies the second statement in the proposition. ∎

7.3 Mod 𝑝 behavior of discriminants

Set-up 7.3.1

Suppose that, in Set-up 7.1.3, B = Spec ⁡ ( O B ) for a local ring O B , so that the vector bundles V 4 and V 6 are trivial O B -modules. By choosing O B -generators for V 4 and V 6 , we identify A ⁢ ( V 4 ⊕ V 6 ) with A d 1 ⊕ A d 2 , where d 1 = 4 ⁢ h − g + 1 and d 2 = 6 ⁢ h − g + 1 . Assume that 2 ⁢ h ≥ g + 1 as in the results in Section 7.2. Consider P 1 = Proj ⁡ O B ⁢ [ u , v ] . Let A 1 = Spec ⁡ ( O B ⁢ [ u ] ) be the v = 1 chart on P 1 , and let ∞ : B → P 1 denote the section defined by v = 0 . Let W r be the O B -module of degree 𝑟 homogeneous polynomials in O B ⁢ [ u , v ] or equivalently the module of polynomials with degree at most 𝑟 in O B ⁢ [ u ] . Consider the open subscheme T ⊆ A ⁢ ( W 4 d 1 ⊕ W 6 d 2 ) consisting of the points of the form

{ ( f 1 , … , f d 1 , g 1 , … , g d 2 ) ∣ the common vanishing locus ⁢ V ⁢ ( { f i , g j } ) = ∅ } .

Then it is clear that there is a natural morphism P B 1 × B T = P T 1 → P ( 4 , 6 ) ⁢ ( V 4 ⊕ V 6 ) . By setting v = 1 in the polynomials f i and g j , we also obtain a 𝐵-morphism

A B 1 × B T = A T 1 → A ⁢ ( V 4 ⊕ V 6 ) ∗ .

Recall that A ⁢ ( V 4 ⊕ V 6 ) ∗ denotes the affine space A ⁢ ( V 4 ⊕ V 6 ) minus the zero section. This morphism fits into a commutative diagram

For the content below, recall Definition 6.1.1 and definition of X ̃ , 𝒳, 𝔇, D ̄ , and 𝔘 in Set-up 7.1.3. Assume that 𝑘 is an algebraically closed field of characteristic ≠ 2 , 3 .

Proposition 7.3.2

Suppose that B = Spec ⁡ ( k ) . Then the family A T 1 / T strongly dominates A ⁢ ( V 4 ⊕ V 6 ) ∗ via 𝜑. Moreover, for a general point t ∈ T , we have φ ̄ t ⁢ ( ∞ ) ∉ D ̄ .

Proof

The first statement is an exercise of dimension counting, so we omit the details. For the second statement, it suffices to exhibit a single such 𝑡 as the condition is open. Note that the automorphism group of P 1 naturally acts on 𝑇. We start with any point t ′ ∈ T such that, for some point w ∈ A 1 , φ t ⁢ ( w ) ∉ D . Then we can always apply an automorphism of P 1 to switch 𝑤 and ∞. This gives us a point t ∈ T and, by construction, φ ̄ t ⁢ ( ∞ ) ∉ D ̄ . ∎

Lemma 7.3.3

Suppose char ⁡ k = p ≥ 5 and, in Set-up 7.3.1, 𝐵 is taken to be Spec ⁡ ( W ) for W : = W ( k ) . Assume that D K has codimension 1 in A ⁢ ( V 4 ⊕ V 6 ) K ∗ over K : = W [ 1 / p ] . Then, for a general 𝑘-point t ∈ T k , φ ̄ t has the following properties:

φ ̄ t ⁢ ( ∞ ) ∉ D ̄ k , the total space of the family φ ̄ t ∗ ⁢ ( X ) → P k 1 is smooth, and every fiber has at most a single ODP singularity;
φ ̄ t ∗ ⁢ ( D ̄ k ) , or equivalently φ t ∗ ⁢ ( D k ) , is reduced;
for every t ̃ ∈ T ⁢ ( W ) lifting 𝑡, φ t ̃ ∗ ⁢ ( D ) is étale over 𝑊, so that the open subcurve
φ t ̃ ∗ ⁢ ( U ) ⊆ A t ̃ 1
has a good compactification relative to 𝑊.

Proof

Using that D ̄ is proper over 𝑊 and D K has codimension 1 in A ⁢ ( V 4 ⊕ V 6 ) K ∗ , it is not hard to see that D k is also of codimension exactly 1 in A ⁢ ( V 4 ⊕ V 6 ) k ∗ . We break the proof into three steps.

Step 1. By Proposition 7.2.2 and Remark 6.1.4, the irreducible components of maximal dimension of D K and ( D k ) red are unique. Let us denote them by D K ∘ and D k ∘ respectively. Moreover, by Proposition 7.2.2, there exists an open dense subscheme V ⊆ D k ∘ such that, for every v ∈ V ⁢ ( k ) , X v has at most a single ODP singularity. In particular, by Proposition 7.3.2, as well as Proposition 6.1.6 and its proof, for a general point t ∈ T ⁢ ( k ) , φ ̄ t ⁢ ( ∞ ) ∉ D ̄ k , the total space of the pullback family φ t ∗ ⁢ ( X ̄ ) is smooth, and the image of φ t only intersects D k ∘ on 𝑉, so that every singular fiber of φ t ∗ ⁢ ( X ̄ ) over A t 1 has a single ODP; moreover, as φ ̄ t ⁢ ( ∞ ) ∉ D ̄ k , the total space of φ ̄ t ∗ ⁢ ( X ) is smooth. Hence we may conclude (a).

Step 2. Next, we show the following claim (⋆): if 𝑡 is a general point, for any t ̃ ∈ T ⁢ ( W ) lifting 𝑡, Z : = φ t ̃ ∗ ( D ) is flat over 𝑊. Let D 0 , … , D r be the irreducible components of 𝔇 such that D 0 is the component which contains D K ∘ . We claim that D 0 contains D k ∘ as well. To simplify notation, let us write A ⁢ ( V 4 ⊕ V 6 ) as 𝒜. Then D 0 , K ⊆ A K ∗ is cut out by a single polynomial 𝐹 in the coordinates of the affine space A K . By minimally clearing denominators, we may assume that the coefficients of 𝐹 are defined in 𝑊 and generate 𝑊. Using the fact that 𝒜 is affine and O A is a UFD, one checks that the Zariski closure of D 0 , K in 𝒜 contains the vanishing locus V ⁢ ( F ) of F ∈ O A . Note that 𝐹 is weighted-homogeneous, so V ⁢ ( F ) k ⊆ A k at least contains the origin. In particular, V ⁢ ( F ) → B is surjective. By [66, Tag 0B2J], we have dim V ⁢ ( F ) k = dim V ⁢ ( F ) K . This implies that dim D 0 , k = dim D 0 , K . By the uniqueness of D k ∘ as an irreducible component of maximal dimension, we conclude that D k ∘ ⊆ D 0 . By applying [66, Tag 0B2J] again, we also conclude that, for any i > 0 , D i , k has codimension at least 2 in A k ∗ .

Set U ⊆ D 0 to be the complement of the closed subscheme ⋃ i > 1 ( D 0 ∩ D i ) . Then ( U k ) red is dense in D k ∘ . As 𝑡 is general and the family A T k 1 strongly dominates A ⁢ ( V 4 ⊕ V 6 ) k , we may assume that the intersection im ⁡ ( φ t ) ∩ ( D k ) red is transverse and lies in U k . Now we can prove claim (⋆). Indeed, note that D 0 is a Weil divisor, and hence also a Cartier divisor of A ∗ , as A ∗ is regular. This implies that Z = φ t ̃ ∗ ⁢ ( D 0 ) is everywhere locally cut out in P W 1 by a single equation. Since Z k ⊆ P k 1 is of codimension 1, 𝑍 is flat over 𝑊 by [66, Tag 00MF].

Step 3. Finally, we show (b) and (c) simultaneously. Note that if we show (c) for some t ̃ , then we can already conclude (b), which conversely implies (c) for all t ̃ . Indeed, O Z is isomorphic to W ⁢ [ x ] / ( f ⁢ ( x ) ) for some f ⁢ ( x ) ∉ p ⁢ W ⁢ [ x ] . If Z k = φ ∗ ⁢ ( D k ) is reduced, then by Hensel’s lemma, 𝑍 has to be a disjoint union of several copies of 𝑊. Hence it suffices to show (c) for some t ̃ , for which we may assume that the generic fiber φ t ̃ ⊗ K intersects D K transversely. What we are using here is that A T 1 ⊗ K ̄ strongly dominates A ⁢ ( V 4 ⊕ V 6 ) K ̄ ∗ and the points on T ⁢ ( W ) which lift 𝑡 are Zariski dense on T K .

Since 𝑍 is finite and flat, deg K ⁡ Z K = deg k ⁡ Z k . As Z K is reduced, deg ⁡ Z K is the number of closed points on Z K ̄ . Therefore, to show that Z k is reduced, it suffices to show that Z k has the same number of closed points, i.e., | Z k ⁢ ( k ) | = | Z K ̄ ⁢ ( K ̄ ) | . Now we compute | Z k ⁢ ( k ) | by topology. Choose a prime ℓ ≠ p . Since we assumed that φ ̄ t ∗ ⁢ ( X ) is smooth over 𝑘, we may apply the Grothendieck–Ogg–Shafarevich formula in the form of [15, Exposé XVI, Proposition 2.1] to the morphism g : φ ̄ t ∗ ⁢ ( X ) → P k 1 and obtain

χ ⁢ ( φ ̄ t ∗ ⁢ ( X ) ) = χ ⁢ ( X η ̄ t ) ⁢ χ ⁢ ( P k 1 ) − ∑ x ∈ Sing ⁢ ( g ) μ ⁢ ( g , x ) ,

where μ ⁢ ( g , x ) denotes the Milnor number at the singularity 𝑥. By our assumption that, for every z ∈ Z k , the fiber X z is singular at exactly an ODP, we have that | Sing ⁢ ( g ) ⁢ ( k ) | = | Z k ⁢ ( k ) | , and for each x ∈ Sing ⁢ ( g ) , μ ⁢ ( g , x ) = 1 (see case 2 in the proof of [15, Exposé XVI, Proposition 1.13]). Therefore, we have that

| Z k ⁢ ( k ) | = χ ⁢ ( X η ̄ t ) ⁢ χ ⁢ ( P k 1 ) − χ ⁢ ( φ ̄ t ∗ ⁢ ( X ) ) .

By considering how singularities might degenerate, we easily see that the fiber of 𝒳 over each point in Z K ̄ ⁢ ( K ̄ ) has at most an ODP singularity. Therefore, by the same computation as above, for u : = t ̃ K ̄ and a geometric generic point η ̄ u of P u 1 , we have

| Z K ̄ ⁢ ( K ̄ ) | = χ ⁢ ( X η ̄ u ) ⁢ χ ⁢ ( P K ̄ 1 ) − χ ⁢ ( φ ̄ u ∗ ⁢ ( X ) ) .

Using the smooth and proper base change theorem for étale cohomology, it is not hard to see that

χ ⁢ ( φ ̄ u ∗ ⁢ ( X ) ) = χ ⁢ ( φ ̄ t ∗ ⁢ ( X ) ) and χ ⁢ ( X η ̄ u ) = χ ⁢ ( X η ̄ t ) .

Hence we conclude that | Z k ⁢ ( k ) | = | Z K ̄ ⁢ ( K ̄ ) | , as desired. ∎

Note that the fact that the Milnor number for ODPs on a pencil of surfaces is equal to 1 fundamentally uses the p ≠ 2 assumption. Of course, it is irrelevant here because we are working with p ≥ 5 , but we remark that, in [55, Theorem 4.2], the non-reducedness of the relevant discriminant scheme modulo 2 can indeed be explained by the fact that an ordinary quadratic singularity has an even Milnor number in characteristic 2 (see [58, Proposition 3.24]).^[12] We also remark that results in Section 7.2 crucially use the 2 ⁢ h ≥ g + 1 assumption, which is indeed satisfied by the case we care about ( g = h = 1 ).

7.4 Proof of Theorem A

In this section, we work with the following set-up.

Set-up 7.4.1

Let M 1 , 1 be the moduli stack of the pair of a genus 1 curve together with a degree 1 line bundle (i.e., an elliptic curve). Let 𝐵 be the Z ⁢ [ 1 / 6 ] -scheme defined by

{ ( a , b ) ∈ Spec ⁢ ( Z ⁢ [ 1 6 ] ⁢ [ a , b ] ) | 4 ⁢ a 3 − 27 ⁢ b 2 ≠ 0 } .

The Weierstrass equation equips 𝐵 with a surjective morphism B → M 1 , 1 . Let ( ϖ : C → B , L ) be the restriction of the universal family over M 1 , 1 to 𝐵. Apply the constructions in Set-up 7.1.3 with this triple ( B , C , L ) and define the objects

V r = ϖ ∗ ⁢ L r ⁢ ( r = 4 , 6 ) , X ̃ → A ⁢ ( V 4 ⊕ V 6 ) ∗ , and D = Disc ⁢ ( X ̃ / A ⁢ ( V 4 ⊕ V 6 ) ∗ )

accordingly. Below, we write A ⁢ ( V 4 ⊕ V 6 ) ∗ as M ̃ , the open subscheme M ̃ − D as 𝖬, and the restriction of X ̃ to 𝖬 as X ∘ .

Remark 7.4.2

For any algebraically closed field 𝑘 of characteristic p ≠ 2 , 3 and elliptic surface 𝑋 over 𝑘 with p g = q = 1 , there exists a point z ∈ M ̃ ⁢ ( k ) such that 𝑋 is the minimal model of X ̃ z . Moreover, there are no reducible fibers in the elliptic fibration of 𝑋 if and only if z ∈ M ⁢ ( k ) . The choice of 𝑧 is unique up to the G m -action on 𝖬 given by

λ ⋅ ( a 4 , a 6 ) = ( λ 4 ⁢ a 4 , λ 6 ⁢ a 6 ) ,

where ( a 4 , a 6 ) is the relative coordinate on A ⁢ ( V 4 ⊕ V 6 ) .

We need a lower bound on the rank of the Kodaira–Spencer map, or equivalently the image of the period morphism over ℂ.

Lemma 7.4.3

For a general z ∈ M C and X : = X z , the Kodaira–Spencer map

T z ⁢ M C → Hom ⁢ ( H 1 ⁢ ( Ω X 1 ) , H 2 ⁢ ( O X ) )

has rank at least 3.

Proof

This follows a construction of Ikeda and the Artin–Brieskorn resolution. In [27], Ikeda constructed a subfamily of elliptic surfaces over ℂ with the given invariants p g = q = 1 using bielliptic curves of genus 3. Let C ̃ be a bielliptic curve of genus 3, equipped with an involution 𝜎 such that C ̃ / σ is a smooth genus 1 curve 𝐶. On the symmetric square C ̃ ( 2 ) of C ̃ , 𝜎 also lifts to an involution σ ( 2 ) . Consider the surface Y ′ = C ̃ ( 2 ) / σ ( 2 ) , which is shown to be a projective surface of Kodaira dimension 1 with 6 ODPs. Its minimal resolution 𝑌 is an elliptic surface with p g = q = 1 . By [27, Proposition 2.9], the morphism C ̃ → C can be recovered from 𝑌. Note however the Weierstrass model of 𝑌 is singular, so 𝑌 is not given by a point on 𝖬.

Note that, for every ℂ-point s ∈ M ̃ C with image 𝑡 in B C , 𝑠 is given by a pair

( a 4 , a 6 ) ∈ H 0 ⁢ ( L t 4 ) × H 0 ⁢ ( L t 6 ) .

Let M C ′ ⊆ M ̃ C be the open subscheme which consists of those 𝑠 such that Δ s = 4 ⁢ a 4 3 − 27 ⁢ a 6 2 does not vanish identically on the base curve C t . Then X ̃ s is the Weierstrass normal form of an elliptic surface and by [30, Theorem 1] has at most rational double point singularities. By applying the Artin–Brieskorn resolution [3] to X ̃ | M C ′ , we obtain a smooth and proper algebraic space X C ♯ → M C ♯ , where M C ♯ is an algebraic space which admits a morphism to M C ′ bijective on geometric points. Moreover, [3, Theorem 2] tells us that, for any ℂ-point 𝑧 of M C ♯ which maps to a point 𝑠 of M C ′ , the Henselianization of M C ♯ at 𝑧 maps surjectively to that of M C ′ at 𝑠. Let M C + be a resolution of singularities of M C ♯ and pull back the family X ♯ | M C ′ to M C + .

Note that all elliptic surfaces which can be constructed as in the first paragraph can be found as fibers of this family over M C + . Let Ω be the period domain parameterizing Hodge structures of K3-type on the integral lattice Λ given by the Betti cohomology of any complex elliptic surface with p g = q = 1 . Let M ̃ C + be the universal cover of M C + . Then, up to an action of O ⁢ ( Λ ) , there is a well defined period map M ̃ C + → Ω . The moduli space of bielliptic curves of genus 3 over ℂ is 4-dimensional, so [27, Theorem 1.1 (1)] implies that the period image of M ̃ C + is of dimension at least 3. Since M C ⊆ M C ′ is open and dense, the discussion on the Henselianization of M C ♯ at ℂ-points in the preceding paragraph implies that the preimage of M C ⊆ M C ′ in M C + is also open and dense. This implies that the preimage of M C in M ̃ C + also has period image of dim ≥ 3 by a continuity argument. ∎

We are now ready to prove (a more general form of) Theorem A.

Theorem 7.4.4

Assume that 𝑘 is a field finitely generated over F p for p ≥ 5 and let k ̄ be a separable closure. Let 𝑋 be an elliptic surface over 𝑘 with p g = q = 1 . If all fibers in the elliptic fibration of X k ̄ are irreducible, then the Tate conjecture holds for 𝑋.

Proof

Let 𝜂 be the generic point of 𝖬. Let 𝔣 and 𝔰 be the classes in NS ⁢ ( X η ∘ ) such that 𝔣 (resp. 𝔰) is given by a smooth fiber (resp. zero section) in the elliptic fibration of X η ∘ . We may polarize the family X ∘ / M with the subspace Q ⁢ ⟨ f , s ⟩ , as it contains the class of ξ η for some relatively ample line bundle 𝝃. Since ⟨ f , f ⟩ = 0 , ⟨ f , s ⟩ = 1 , and ⟨ s , s ⟩ = − 1 , one deduces that, for any s ∈ M ⁢ ( C ) , the lattice PH 2 ⁢ ( X s ∘ , Z ( p ) ) tf is self-dual for every p ≥ 5 . Note that M C is clearly connected, and by Lemma 7.4.3 and Proposition 2.2.5, the family ( X ∘ / M , ξ ) | M C has maximal monodromy, as defined in Definition 2.2.3. By Remark 7.4.2, it suffices to apply Proposition 5.3.3 (a) to the family f : X ∘ → M , for which we only need to prove that L : = R 2 f ∗ Q 2 ( 1 ) has constant λ geo over Spec ⁡ ( Z ⁢ [ 1 / 6 ] ) .

Let 𝖵 be the VHS on M C given by R 2 ⁢ f C ⁣ ∗ ⁢ Q ⁢ ( 1 ) . For convenience, we say that an irreducible smooth ℂ-variety 𝑍 admitting an understood morphism to M C is admissible if its image is not contained in the Noether–Lefschetz loci of 𝖵 and the restriction V | Z is non-isotrivial. Let 𝑝 be a prime at least 5. By Lemma 5.3.4, it suffices to find a smooth connected 𝑊-curve 𝐶 with a morphism to 𝖬 such that C C → M C is admissible and 𝐶 has a good compactification over 𝑊.

We construct 𝐶 in two steps. First, let b ∈ B ⁢ ( k ) be any point and B ̂ b the formal completion of B W at 𝑏. We claim that, for some b ̃ ∈ B ̂ b ⁢ ( W ) , M b ̃ ⊗ C is admissible. Indeed, we first observe that, since dim B C = 2 , Lemma 7.4.3 implies that, for a general point z ∈ B C , V | M z is non-isotrivial. Note that the 𝐾-points given by an analytically dense subset of B ̂ b ⁢ ( W ) are Zariski dense on B K (and hence also on B K ̄ ). As the Noether–Lefschetz (NL) loci of 𝖵 are a countable union of proper closed subvarieties, we may now apply Lemma 6.2.1 to the case 𝑆 being 𝖬 and 𝑁 being the NL loci to conclude that the desired lifting b ̃ exists.

Next, we take the base 𝐵 in the context of Set-up 7.3.1 to be b ̃ above (and hence A ⁢ ( V 4 ⊕ V 6 ) ∗ and 𝔘 in Set-up 7.3.1 are M ̃ b ̃ and M b ̃ respectively in the current context). Then we obtain an irreducible smooth 𝑊-scheme 𝑇 and a morphism φ : A T 1 → M ̃ b ̃ such that φ C defines a strongly dominating families of curves on M ̃ b ̃ ⊗ C by Proposition 7.3.2. Let 𝑡 be a general 𝑘-point on T k such that the conclusion of Lemma 7.3.3 holds.^[13] Since V | M b ̃ ⊗ C is non-isotrivial, for a general z ∈ T C , the restriction of 𝖵 to φ z ∗ ⁢ ( M b ̃ ⊗ C ) is also non-isotrivial. Similarly, applying Lemma 6.2.2 to the NL loci on M b ̃ ⊗ C again, we conclude that, for some t ̃ ∈ T ⁢ ( W ) lifting 𝑡, φ t ̃ ⊗ C ∗ ⁢ ( M b ̃ ⊗ C ) is admissible. Therefore, if we set 𝐶 to be φ t ̃ ∗ ⁢ ( M b ̃ ) , then C C → M C is admissible, and Lemma 7.3.3 (c) guarantees that this 𝐶 admits a good compactification, as desired. ∎

Remark 7.4.5

The bound in Lemma 7.4.3 is now superseded by a recent result of Engel–Greer–Ward [18, Theorem 1.1], which tells us that the period morphism over ℂ is dominant (or equivalently the Kodaira–Spencer map at a general ℂ-point has rank 10). Nevertheless, we retain Lemma 7.4.3 to demonstrate that the proof of Theorem 7.4.4 requires much weaker inputs – a feature that may be useful for applications to other varieties (see Remark 1.4). Also, as can be seen from [21], it is in general highly nontrivial to translate properties of the complex period morphism to its mod ⁡ p reduction when one seeks effective results, so for the current method, the arguments about finding curves in Sections 6 and 7 remain necessary.

We note that [18] has the following consequence.

Proposition 7.4.6

For some finite field 𝑘 of characteristic p ≥ 5 , the highest analytic (or Mordell–Weil) rank of the elliptic curves considered in Theorem A is 10.

Proof

We give a sketch which allows the reader to easily fill in the details. Denote by ρ : M → S the period morphism for 𝖬, where 𝒮 is a suitable orthogonal Shimura variety.^[14] Then [18] tells us that the complex fiber ρ C is dominant. Since the set of CM points on S ⁢ ( C ) is dense, we can find some x ∈ M ⁢ ( C ) such that ρ ⁢ ( x ) is a CM point. Let X C be the corresponding elliptic surface. That ρ ⁢ ( x ) is a CM point implies that 𝑋 has CM, i.e., MT ⁢ ( H 2 ⁢ ( X C , Q ) ) is a torus, or equivalently, the endomorphism algebra of the Hodge structure T ( X ) : = NS ( X ) ⟂ ⊆ H 2 ( X C , Q ( 1 ) ) is a CM field 𝐸 and dim E T ⁢ ( X C ) = 1 by Zarhin’s result in [65] (cf. [26, Chapter 3, Theorem 3.9]).

Thanks to Theorem 4.1.9, the fundamental theorem of CM for K3 surfaces [54, Corollary 3.9.2] applies verbatim to our elliptic surfaces. This allows us to apply [28, Theorem 1.1] up to the obvious adaptations. Moreover, X C is defined over some number field K ⊇ E . Let X K be a 𝐾-model and 𝐹 the maximal totally real subfield of 𝐸. We can always find some place 𝔮 of 𝐹 with sufficiently large residue characteristic and a place 𝑣 of 𝐾 above 𝔮 such that 𝔮 does not split in 𝐸 and X K has good reduction at 𝑣 (or more precisely, x ∈ im ⁡ ( M ⁢ ( O K , ( v ) ) → M ⁢ ( C ) ) , where O K , ( v ) is the localization of the ring of integers O K at 𝑣). By Ito’s theorem, the mod ⁡ v reduction X k ⁢ ( v ) is supersingular, where k ⁢ ( v ) is the residue field.

Let 𝑘 be a finite extension of k ⁢ ( v ) and set X = X k ⁢ ( v ) ⊗ k . Let π : X → C be the elliptic fibration of 𝑋 and let ℰ be the generic fiber over k ⁢ ( C ) . Over the algebraic closure k ̄ , rank ⁡ NS ⁢ ( X k ̄ ) = rank ⁡ H ét 2 ⁢ ( X k ̄ , Q ℓ ⁢ ( 1 ) ) = 12 . As X k is given by a 𝑘-point on 𝖬, all singular fibers in its elliptic fibration are geometrically irreducible. Therefore, by the Shioda–Tate formula [56, Corollary 6.13], up to replacing 𝑘 by a finite extension, rank ⁡ E ⁢ ( k ⁢ ( C ) ) = 12 − 2 = 10 . ∎

8 Surfaces with p g = K 2 = 1 and q = 0

Recall our notation for weighted projective spaces in Section 7.1.2 and set

Q : = ( 1 , 2 , 2 , 3 , 3 ) .

Throughout this section, 𝑘 denotes an arbitrary algebraically closed field of characteristic ≠ 2 , 3 unless otherwise specified.

Theorem 8.1

Let 𝑋 be a minimal surface over 𝑘 with p g = K X 2 = 1 and q = 0 . Then the canonical model X ′ of 𝑋 only has rational double point singularities. Moreover, if we let ( x 0 , x 1 , x 2 , x 3 , x 4 ) be the coordinates of P Q ⁢ ( k ⊕ 5 ) , then for j = 1 , 2 , there exist linear forms α j and cubic forms F j such that X ′ is isomorphic to the subvariety of P Q ⁢ ( k ⊕ 5 ) defined by the two equations

(8.1) x 3 2 + x 0 ⁢ x 4 ⁢ α j ⁢ ( x 0 2 , x 1 , x 2 ) + F j ⁢ ( x 0 2 , x 1 , x 2 ) = 0 .

Proof

When k = C (and hence also any 𝑘 with char ⁡ k = 0 ), these surfaces are classified in [8, 59], and the statements above are taken from [8, Theorem 1.7, Proposition 1.8], where [8, Proposition 1.8] is a consequence of [8, Theorem 1.7], and the only ingredient in [8, Theorem 1.7] (via the proof of [8, Theorem 1.4]) that requires characteristic 0 is [6, Theorem 2], which states that the linear series | 4 ⁢ K X | is base point free. This is shown to hold in positive characteristic by [17, Main Theorem (ii)]. Also, in [8], it is shown that, over ℂ, the condition q = 0 follows from p g = K X 2 = 1 , but here we put q = 0 as part of the assumption. ∎

From now on, we set U : = Z [ 1 / 6 ] and P : = P Q ( O U ⊕ 5 ) , and view 𝑘 as a geometric point on 𝑈. In particular, P k = P Q ⁢ ( k ⊕ 5 ) .

Lemma 8.2

The line bundle O P k ⁢ ( 6 ) on P k is very ample.

Proof

By [5, Theorem 4B.7 (c)], it suffices to show that, for any positive integer ℎ, and every tuple of natural numbers ( A , B , C ) such that A + 2 ⁢ B + 3 ⁢ C = 6 + 6 ⁢ h , there exists a tuple of natural numbers ( a , b , c ) such that a ≤ A , b ≤ B , c ≤ C and a + 2 ⁢ b + 3 ⁢ c = 6 ⁢ h . One easily checks this with an elementary inductive argument. ∎

Proposition 8.3

For every smooth hypersurface Y ⊆ P k , the embedding

ι : Y ↪ | O P k ⁢ ( 6 ) |

is Lefschetz in the sense of [15, Exposé XVII, §2.3].

Note that, unlike P k , the complete linear system | O P k ⁢ ( 6 ) | is a usual projective space (i.e., the weight is ( 1 , 1 , … , 1 ) ).

Proof

We slightly adapt the proof of [15, Exposé XVII, Theorem 2.5.1] to take care of the weights of variables. Since the coordinate x 0 has degree 1, the x 0 ≠ 0 locus of P k is an affine chart isomorphic to A k 4 with coordinates y i : = x i / x 0 deg ⁡ x i . We view the affine variable y i as having degree deg ⁡ x i . Let 𝑃 be a point on 𝑌 and assume that the projection of A 4 k to the y 4 -axis is étale at 𝑃. Up to a translation, we further assume that y i = 0 at 𝑃 for i = 1 , 2 , 3 . By [15, Exposé XVII, Proposition 3.3 and Corollary 3.5.0], it suffices to show that, for some H ∈ | O P k ⁢ ( 6 ) | , the intersection H ∩ Y has an ODP at 𝑃. To this end, it suffices to present a polynomial in variables y 1 , y 2 , y 3 of weighted degree at most 6 such that its vanishing locus has an ODP at the origin, for which one may simply take y 3 2 + y 1 ⁢ y 2 = 0 . ∎

We now parameterize the surfaces in Theorem C.

Set-up 8.4

Let B ⊆ A U 13 (resp. B ′ ) be the open subscheme whose geometric points are pairs ( α , F ) such that the degree 6 hypersurface defined by (8.1) has isolated singularities (resp. is non-singular). Let M ̃ be the trivial projective bundle | O P ⁢ ( 6 ) | × U B over 𝐵. Let X ̃ → M ̃ be the natural family whose fiber over a point ( H , H 0 ) ∈ M ̃ is given by H ∩ H 0 . Set D : = Disc ( X ̃ / M ̃ ) (see Definition 6.1.3), M : = M ̃ − D , and X ∘ : = X ̃ | M . Then every surface of general type with ample canonical bundle, p g = K 2 = 1 , and q = 0 defined over an algebraically closed field of characteristic p > 3 can be found as a geometric fiber of X ∘ on 𝖬.

Proposition 8.5

For every geometric point s → B ′ , the fiber D s is generically reduced.

Proof

Let H 0 ⊆ P k ⁢ ( s ) be the smooth degree 6 hypersurface defined by 𝑠. By Proposition 8.3, H 0 ↪ | O P s ⁢ ( 6 ) | is a Lefschetz embedding. In particular, the codimension of D s in | O P s ⁢ ( 6 ) | is 1. This implies that 𝔇 is flat over B ′ , and hence the degree of D s in the projective space | O P s ⁢ ( 6 ) | is independent of 𝑠.

Therefore, it suffices to show that d s : = deg ( D s ) red is independent of 𝑠, because we know D s is reduced if 𝑠 is a geometric generic point over B ′ . The point then is that we can compute deg ( D s ) red by topology and this computation is independent of 𝑠. Let γ : P k 1 ↪ | O P s ⁢ ( 6 ) | be a general pencil, which by Proposition 8.3 is necessarily a Lefschetz pencil for H 0 . Then d s is simply the number of singular fibers on X ̃ | γ . Note that the total space of X ̃ | γ is smooth and every singular fiber is smooth away from an ODP. Let η ̄ be the geometric generic point of P k ⁢ ( s ) 1 . We apply the key argument in Step 3 of Lemma 7.3.3 again: by the Leray spectral sequence and the Grothendieck–Ogg–Shafarevich formula, we have

d s = χ ⁢ ( X ̄ | γ ) − χ ⁢ ( P 1 ) ⁢ χ ⁢ ( X η ̄ ) .

Again, this uses the fact that char ⁡ k ⁢ ( s ) ≠ 2 . By a simple deformation argument, one sees that the right-hand side of the above equation is independent of 𝑠, and hence so is the left-hand side. ∎

Now we are ready to prove Theorem C. We recall that, over ℂ, a minimal surface 𝑋 with the given invariants is known to be simply connected [59]; hence H 2 ⁢ ( X , Z ) is torsion-free.

Proof

Let us polarize the family f : X ∘ → M with the relative canonical bundle 𝒦. Then, for each s ∈ M ⁢ ( C ) , PH 2 ⁢ ( X s , Z ) is unimodular, as H 2 ⁢ ( X s , Z ) is unimodular and K s 2 = 1 . Let 𝖵 be the VHS on R 2 ⁢ f C ⁣ ∗ ⁢ Q ⁢ ( 1 ) . Then, since 𝖵 over M C has period dimension 18 (see [8, 59]), we know that the family ( X ∘ / M , K ) | M C has maximal monodromy by Proposition 2.2.5.

The proof is entirely similar to the proof of Theorem 7.4.4, so we only explain the adaptations. Again, we apply Proposition 5.3.3 to the family X ∘ / M together with Lemma 5.3.4 for L : = R 2 f ∗ Q 2 ( 1 ) and any p ≥ 5 . Set W : = W ( F ̄ p ) , K : = W [ 1 / p ] , choose K ̄ ≅ C , and call a smooth irreducible ℂ-variety 𝑍 admitting an understood morphism to M C admissible if V | Z is non-isotrivial, and the image of 𝑍 is not contained in the Noether–Lefschetz (NL) loci. The task is to find a smooth connected 𝑊-curve 𝐶 with a morphism to 𝖬 such that C C → M C is admissible and 𝐶 has a good compactification over 𝑊.

Using that 𝖵 over M C has period dimension 18 > dim B C , for a general point on B C , the restriction of 𝖵 is not isotrivial. Now let b ∈ B k ′ be a closed point and B ̂ b ′ the formal completion of B W ′ at 𝑏. Then the 𝐾-points given by any analytically dense subset of B ̂ b ⁢ ( W ) are Zariski dense on B K . By applying Lemma 6.2.1 to the NL loci of 𝖵, we can find a lifting b ̃ ∈ B ′ ⁢ ( W ) such that M b ̃ ⊗ C is admissible. Next, note that M ̃ b ̃ = | O P b ̃ ⁢ ( 6 ) | over 𝑊. By Proposition 8.5, D b ⊆ M ̃ b is generically reduced. By considering the relative Grassmannian of lines on M ̃ b ̃ and applying Lemma 6.2.2, one finds a line L ≅ P W 1 in M ̃ b ̃ such that L ∩ D b ̃ is étale over 𝑊 and ( L − D b ̃ ∩ L ) C ⊆ M C is admissible. Hence we may take C = L − D b ̃ ∩ L . ∎

Funding source: European Research Council

Award Identifier / Grant number: 770936

Funding source: Simons Foundation

Award Identifier / Grant number: 636187

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2101789

Award Identifier / Grant number: DMS-2052665

Funding statement: Paul Hamacher is partially supported by ERC Consolidator Grant 770936: NewtonStrat. Xiaolei Zhao is partially supported by the Simons Collaborative Grant 636187, NSF grant DMS-2101789, and NSF FRG grant DMS-2052665.

Acknowledgements

We wish to thank Jordan Ellenberg, Philip Engel, Shizhang Li, Yuchen Fu, Davesh Maulik, Martin Orr, Alex Petrov, Ananth Shankar, Mark Shusterman, Yunqing Tang, Lenny Taelman, Daichi Takeuchi, Kai Xu, Ruijie Yang for helpful conversations or email correspondences. A special thank goes to Dori Bejleri for generously helping us with lots of technical questions about elliptic surfaces. We also thank the anonymous reviewer for many helpful comments that improved the exposition of the paper. Ziquan Yang benefited from several extended research visits to discuss this work and is thankful to Shanghai Center of Mathematics, University of Amsterdam, and the Max Planck Institute for their hospitality. Finally, Ziquan Yang wishes to thank Davesh Maulik, Ben Moonen, Ananth Shankar, and his PhD advisor Mark Kisin for their kind encouragements.

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Received: 2025-01-23

Revised: 2025-05-29

Published Online: 2025-07-10

Published in Print: 2025-10-01

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On the Tate conjecture for divisors on varieties with ℎ2,0 = 1 in positive characteristics

Article

Abstract

1 Introduction

Sketch of proofs

Organization of paper

2 Preliminaries

2.1 Motives and norm functors

Proof

2.2 Motives of varieties with h 2 , 0 = 1

Theorem 2.2.2 ([65, §2])

Proof

Proof

Proof

Proof

3 Moduli interpretations of Shimura varieties

3.1 Systems of realizations

Lemma 3.1.6 ([63, (3.4.1)])

Proof

3.2 Shimura varieties

Theorem 3.2.5 ([63, (4.3.1)])

3.3 Orthogonal Shimura varieties over totally real fields

Proof

Proof

3.4 Integral model

Proof

4 Period morphisms

4.1 The basic set-up

Proof

Proof

4.2 Case (R+): Maximal monodromy

Proof

Proof

Proof

Proof

Proof of Theorem 4.2.6 for 𝑚 odd

Proof

Proof of Theorem 4.2.6 for 𝑚 even

4.3 Case (R2 ′ ): Non-maximal monodromy

Proof

Proof of Theorem 4.1.10

5 Proof of Theorem B

5.1 A specialization lemma for monodromy

Proof

Proof

5.2 An effective theorem

Proof

Proof

Proof

5.3 Proof of Theorem B

Proof

Proof

Proof

6 Deforming curves on parameter spaces

6.1 Families of curves which homogeneously dominate a variety

Proof

Proof

6.2 Applications of the Baire category theorem

Proof

Proof

7 Elliptic surfaces with p g = q = 1

7.1 Generalities on elliptic surfaces

Proof

Proof

7.2 Nonlinear Bertini theorems for families of elliptic surfaces

Proof

Proof

7.3 Mod 𝑝 behavior of discriminants

Proof

Proof

7.4 Proof of Theorem A

Proof

Proof

Proof

8 Surfaces with p g = K 2 = 1 and q = 0

Proof

Proof

Proof

Proof

Proof

On the Tate conjecture for divisors on varieties with ℎ^2,0 = 1 in positive characteristics

4.3 Case (R2^′): Non-maximal monodromy