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Eisenstein degeneration of Euler systems

  • David Loeffler ORCID logo and Oscar Rivero Salgado ORCID logo EMAIL logo
Published/Copyright: August 21, 2024
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 814

Abstract

We discuss the theory of Coleman families interpolating critical-slope Eisenstein series. We apply it to study degeneration phenomena at the level of Euler systems. In particular, this allows us to prove relations between Kato elements, Beilinson–Flach classes and diagonal cycles, and also between Heegner cycles and elliptic units. We expect that this method could be extended to construct new instances of Euler systems.

Introduction

Overview

In the beautiful survey article [6], Bertolini et al. described several families of global cohomology classes arising from modular curves, including the Gross–Kudla–Schoen diagonal-cycle classes for a triple product of modular forms, and the Euler systems of Beilinson–Flach and Beilinson–Kato elements for two and for one modular form. They noted that the latter two constructions formally behave, in many ways, as if they were a “degenerate case” of the Gross–Kudla–Schoen classes with one or more of the cusp forms replaced by Eisenstein series. However, while this formal resemblance has proved very informative as a heuristic to guide the development of the theory, making it into a rigorous mathematical statement is not straightforward: since the Gross–Kudla–Schoen cycle extends to the triple product of the compactified modular curves X 1 ⁢ ( N ) , the projection of this cycle to a non-cuspidal Hecke eigenspace is zero.

In this work, we develop an approach which allows us to make this “Eisenstein degeneration” of Euler systems into a rigorous theory. Our approach is based on two ingredients: the theory developed in [31] to study the variation of Euler systems in Coleman families; and the existence of critical-slope Eisenstein series, which are Eisenstein series arising as specialisations of Coleman families which are generically cuspidal.

Using this method, we show the following: if 𝐟 is a Coleman family passing through a critical-slope Eisenstein series f β , and g , h are (cuspidal) Coleman families, then the process of “specialisation at f β ” relates Euler systems in the following way.

  1. The Beilinson–Flach classes for f × g go to a multiple of the Beilinson–Kato classes for 𝐠. This setting is studied in Section C1.

  2. The triple-product classes for f × g × h are sent to the Beilinson–Flach classes for g × h . This scenario is presented in Section C2.

  3. The Heegner classes for f × λ , where 𝜆 is a (suitable) Grössencharacter of an imaginary quadratic field, go to the elliptic-unit classes for 𝜆. This is the content of the final Section C3.

More precisely, in each case, we show that the image of the Euler system for the “larger” family is an Euler system class for the “smaller” family, multiplied by an additional, purely local “logarithm” term (and also by an extra 𝑝-adic 𝐿-function factor in case (ii), which can be naturally interpreted in terms of the Artin formalism for 𝐿-functions).

Besides the intrinsic interest in relating these natural and important objects to each other, we hope that the techniques that we develop here may be of use in constructing new Euler systems, as in forthcoming work of Barrera et al. discussed below. We hope to pursue this further in a future work.

We have also included an extensive section (Part B) devoted to recalling the main points in the theory for each of the Euler systems discussed in this note. We hope that this could help to reconcile the notation used in different papers, and could help as a guide to the less experienced reader willing to gain expertise in the area. Some of the results we present have been proved under certain simplifying assumptions, like considering tame level 1 for the degeneration of diagonal cycles to Beilinson–Flach classes, or restricting to class number 1 when discussing how to recover elliptic units beginning with Heegner points. Our purpose was illustrating our method without dealing with certain technical difficulties, but of course most of these hypotheses can be removed with some extra work.

Relation to other work

There are a number of prior works studying the specialisation of families of 𝑝-adic 𝐿-functions and/or Euler systems at points of the eigencurve corresponding to cuspidal points of critical slope; see for instance [29] and rather more recently [13].

In a slightly different direction, one can consider families of cusp forms degenerating to weight one Eisenstein series, which exist when the Eisenstein series is non-𝑝-regular. This approach has the advantage that the resulting families are ordinary, but on the other hand, the eigencurve is not smooth (or even Gorenstein) at such points; its local geometry has been studied in detail by Pozzi [35]. A forthcoming work of Barrera, Cauchi, Molina and Rotger will use this approach, applied to diagonal cycles on triple products of quaternionic Shimura curves, in order to define global Galois cohomology classes associated to products of one or two Hilbert modular forms. (This could potentially compensate for the fact that the original construction of the Beilinson–Flach and Beilinson–Kato Euler systems does not generalise to the Hilbert case, owing to the lack of suitable Eisenstein classes.) It will be interesting to explore the relation between their techniques and ours once their manuscript becomes available.

Finally, a recent work of Bertolini, Darmon and Venerucci [10] proves a striking conjecture of Perrin-Riou relying on a comparison between Beilinson–Flach elements associated to weight 1 Eisenstein series and Beilinson–Kato elements, showing the strength of this kind of techniques.

A Critical-slope Eisenstein series and their Galois modules

A1 Notation

We introduce notation for Eisenstein series, following the conventions of [5]. Let ψ , τ be two primitive Dirichlet characters of conductors N 1 , N 2 , and let N = N 1 ⁢ N 2 . Let r ⩾ 0 be such[1] that ψ ⁢ ( − 1 ) ⁢ τ ⁢ ( − 1 ) = ( − 1 ) r . If r = 0 , assume 𝜓 and 𝜏 are not both trivial.

Definition A1.1

We write f = E r + 2 ⁢ ( ψ , τ ) for the (classical) modular form of level 𝑁 and weight r + 2 given by

E r + 2 ⁢ ( ψ , τ ) = ( ∗ ) + ∑ n ⩾ 1 q n ⁢ ∑ n = d 1 ⁢ d 2 ( N 1 , d 1 ) = ( N 2 , d 2 ) = 1 ψ ⁢ ( d 1 ) ⁢ τ ⁢ ( d 2 ) ⁢ d 2 r + 1 ,

where ( ∗ ) is the appropriate constant defined e.g. in [5, §1]. In particular,

a ℓ ⁢ ( f ) = ψ ⁢ ( ℓ ) + ℓ r + 1 ⁢ τ ⁢ ( ℓ )

for primes ℓ ∤ N .

We have two eigenforms of level Γ 1 ⁢ ( N ) ∩ Γ 0 ⁢ ( p ) associated to E r + 2 ⁢ ( ψ , τ ) : one ordinary and one critical-slope, with eigenvalues

α : = ψ ( p ) and β : = p r + 1 τ ( p )

respectively. We denote these two eigenforms by f α and f β respectively.

We shall want to study the 𝑝-adic Galois representations attached to these forms, for a prime p ∤ N . It will be helpful to make the following definition.

Definition A1.2

We say the Eisenstein series f = E r + 2 ⁢ ( ψ , τ ) is 𝑝-decent if one of the following conditions holds:

  • r > 0 ;

  • r = 0 , and for every prime ℓ ∣ N ⁢ p , either the conductor of ψ / τ is divisible by ℓ, or ( ψ / τ ) ⁢ ( ℓ ) ≠ 1 .

Remark A1.3

This is slightly stronger than Bellaïche’s definition of “decent” [3, Definition 1], since we also make the assumption at 𝑝.

A2 Galois representations

We first fix notation for Galois characters. Let 𝑝 be a prime (which will be fixed for the duration of this paper).

Notation

Let ℓ be a prime. We write Frob ℓ for an arithmetic Frobenius element at ℓ in Gal ⁡ ( Q ̄ / Q ) ; this depends, of course, on the choice of an embedding of Q ̄ into Q ̄ ℓ , and is well-defined modulo the inertia subgroup for this embedding.

The 𝑝-adic cyclotomic character Gal ⁡ ( Q ̄ / Q ) → Q p × will be denoted by 𝜀, so that

ε ⁢ ( Frob ℓ ) = ℓ for ⁢ ℓ ≠ p .

If 𝜒 is a Dirichlet character, we interpret 𝜒 as a character Gal ⁡ ( Q ̄ / Q ) → C × unramified at all primes ℓ not dividing the conductor, and mapping Frob ℓ − 1 to χ ⁢ ( ℓ ) for all such primes ℓ.

Now let χ , ψ be Dirichlet characters, and r ⩾ 0 an integer, as in Definition A1.1, and choose an embedding Q ⁢ ( ψ , τ ) ↪ L , where Q ⁢ ( ψ , τ ) ⊂ C is the finite extension of 𝐐 generated by the values of 𝜓 and 𝜏, and 𝐿 is a finite extension of Q p . Hence we may regard the 𝑞-expansion coefficients a n ⁢ ( f ) of f = E r + 2 ⁢ ( ψ , τ ) as elements of 𝐿.

Theorem A2.1

Theorem A2.1 (Soulé)

If 𝑓 is a 𝑝-decent Eisenstein series, there are exactly three isomorphism classes of continuous Galois representations ρ : G Q → GL 2 ⁡ ( L ) which are unramified at primes ℓ ∤ N ⁢ p and satisfy tr ⁡ ρ ⁢ ( Frob ℓ − 1 ) = a ℓ ⁢ ( f ) . These are as follows.

  1. The semisimple representation ψ ⊕ τ ⁢ ε − 1 − r .

  2. Exactly one non-split representation having τ ⁢ ε − 1 − r as a subrepresentation. This representation splits locally at ℓ for every ℓ ≠ p , but does not split at 𝑝, and is not crystalline (or even de Rham).

  3. Exactly one non-split representation having 𝜓 as a subrepresentation. This representation splits locally at ℓ for every ℓ ≠ p , and is crystalline at 𝑝.

This follows from cases of the Bloch–Kato conjecture due to Soulé; see [4, §5.1] for the statement in this form. If 𝑓 is not 𝑝-decent, there will be extra representations in case (3), but we can repair the statement by only considering representations which are assumed to be unramified (resp. crystalline) at each prime ℓ ≠ p (resp. at ℓ = p ) where the decency hypothesis fails.

Remark A2.2

The dual of the representation in (3) is the one that Bellaïche describes as the “preferred representation” associated to 𝑓 (see [3, Lemma 2.10]).

A3 Classical cohomology

For an 𝐿-vector space 𝑀 with an action of the Hecke algebra of level 𝑁 and weight r + 2 , we let M ⁢ [ T = f ] signify the maximal subspace of 𝑀 on which the Hecke operators T ⁢ ( ℓ ) , U ⁢ ( ℓ ) act as a ℓ ⁢ ( f ) . Dually, we let M ⁢ [ T ′ = f ] for the corresponding eigenspace for the dual Hecke operators T ′ ⁢ ( ℓ ) , U ′ ⁢ ( ℓ ) .

Let Y 1 ⁢ ( N ) denote the modular curve classifying[2] elliptic curves with a point of order 𝑁 (identified with a quotient of the upper half-plane via τ ↔ ( C / ( Z + Z ⁢ τ ) , 1 N ) , so the cusp ∞ is not defined over 𝐐). We let Y 1 ⁢ ( N ) ̄ denote the modular curve over Q ̄ .

A3.1 Étale cohomology

Let H Q p be the relative Tate module of the universal elliptic curve over Y 1 ⁢ ( N ) , and H Q p ∨ for its dual (the relative étale H 1 ). We write

V r = Sym r ⁡ ( H Q p ∨ ) ⊗ Q p L .

Proposition A3.1

Let f = E r + 2 ⁢ ( ψ , τ ) as above. Then the eigenspaces

H c 1 ⁢ ( Y 1 ⁢ ( N ) ̄ , V r ) ⁢ [ T = f ] and H 1 ⁢ ( Y 1 ⁢ ( N ) ̄ , V r ) ⁢ [ T = f ]

are both one-dimensional over 𝐿, but the natural map between them is 0. The group Gal ⁡ ( Q ̄ / Q ) acts on the former by 𝜓, and on the latter by τ ⁢ ε − 1 − r .

Note that H c 1 ⁢ ( Y 1 ⁢ ( N ) ̄ , V r ) is canonically isomorphic to the space Symb Γ 1 ⁢ ( N ) ⁡ ( V r ) of modular symbols. We endow these spaces with Galois actions following the normalisations of [24].

Remark A3.2

If we work instead with the modular curve 𝑌 of level

Γ = Γ 1 ⁢ ( N ) ∩ Γ 0 ⁢ ( p ) ,

and use either of the two 𝑝-stabilised eigenforms f α , f β , then the resulting spaces are isomorphic to those at level 𝑁 via degeneracy maps, so the Galois actions are the same as above. That is, for ? = α or 𝛽, the spaces

H c 1 ⁢ ( Y ̄ , V r ) ⁢ [ T = f ? ] and H 1 ⁢ ( Y ̄ , V r ) ⁢ [ T = f ? ]

are 1-dimensional, and isomorphic as Galois representations to 𝜓 and τ ⁢ ε − 1 − r respectively.

A3.2 De Rham cohomology

We have similar results for de Rham cohomology: the eigenspaces

H dR , c 1 ⁢ ( Y 1 ⁢ ( N ) , V r ) ⁢ [ T = f ] and H dR 1 ⁢ ( Y 1 ⁢ ( N ) , V r ) ⁢ [ T = f ]

are both 1-dimensional over 𝐿, but the former has its grading concentrated in degree 0, and the latter in degree r + 1 . Using the BGG complex (with and without compact supports; see [27, Proposition 2.4.1]), we can define classical coherent-cohomology eigenclasses

η f ∈ H 1 ⁢ ( X 1 ⁢ ( N ) Q ̄ , ω − r ⁢ ( − cusps ) ) ⁢ [ T = f ] and ω f ∈ H 1 ⁢ ( X 1 ⁢ ( N ) Q ̄ , ω r + 2 ) ⁢ [ T = f ]

whose images span the 𝑓-eigenspaces in H dR , c 1 and H dR 1 respectively; these are characterised, as usual, by the requirement that ω f be the image of the differential form associated to 𝑓, and η f pair to 1 with ω f ∗ under Serre duality. Here, f ∗ is the eigenform conjugate to 𝑓.

In general, η f and ω f will not be defined over 𝐿, only over L ⁢ ( μ N ) . We can correct this by multiplying them by a suitable Gauss sum, to make them 𝐿-rational, giving modified classes η ̃ f , ω ̃ f spanning the corresponding spaces over 𝐿, exactly as in the cuspidal case considered in [25, §6.1]. However, for our purposes, it is simpler to assume that 𝐿 is large enough that it contains an 𝑁-th root of unity (so the Gauss sum is in 𝐿 anyway), in which case η f and ω f are 𝐿-rational. We shall assume that 𝐿 satisfies this condition henceforth.

A4 Overconvergent cohomology

Let Γ = Γ 1 ⁢ ( N ) ∩ Γ 0 ⁢ ( p ) , and Y = Y ⁢ ( Γ ) , equipped with a model over 𝐐 compatibly with the one above for Y 1 ⁢ ( N ) (so 𝑌 is the modular curve denoted Y ⁢ ( 1 , N ⁢ ( p ) ) in [21]). For each integer r ⩾ 0 , we have an étale sheaf D r on 𝑌, corresponding to the representation of Γ on the dual space of the Tate algebra in one variable 𝑧, with the action of Γ twisted by ( a + c ⁢ z ) r . This also makes sense for r < 0 (and indeed for any locally analytic character of Z p × ).

Remark A4.1

This sheaf is the sheaf denoted D r , m ⁢ ( H 0 ) ⁢ ( − r ) in [31], with 𝑚 an auxiliary parameter (radius of analyticity); we take m = 0 and drop it from the notation.

These sheaves are normally considered as topological (Betti) sheaves, but they can be promoted to étale sheaves on the canonical model of 𝑌 as a Z ⁢ [ 1 / N ⁢ p ] -scheme; cf. [2, 31]. Note that the Hecke operators away from 𝑝 act on the cohomology of D r , as does the operator U ⁢ ( p ) , but U ′ ⁢ ( p ) does not act on this sheaf.

As in [3, §3.2], we have a specialisation morphism D r → ρ r V r , which fits into an exact sequence of sheaves on 𝑌,

0 → D − 2 − r ⁢ ( − 1 − r ) → θ r + 1 D r → ρ r V r → 0 ,

where ρ r is the natural specialisation map, ( − 1 − r ) denotes a Tate twist, and θ r + 1 is the dual of ( r + 1 ) -fold differentiation.

A4.1 Compact support

We have H c 2 ⁢ ( Y ̄ , D − 2 − r ) = 0 by a theorem of Stevens [34, Lemma 5.2], and of course, H c 0 ⁢ ( Y ̄ , V r ) is also zero, so we obtain a short exact sequence of compactly supported étale cohomology spaces

(A4.1) 0 → H c 1 ⁢ ( Y ̄ , D − 2 − r ⁢ ( − 1 − r ) ) → θ r + 1 H c 1 ⁢ ( Y ̄ , D r ) → ρ r H c 1 ⁢ ( Y ̄ , V r ) → 0 .

This is an exact sequence of étale sheaves on Spec ⁡ Z ⁢ [ 1 / N ⁢ p ] (equivalently, of Gal ⁡ ( Q ̄ / Q ) -representations unramified outside N ⁢ p ). The map ρ r is compatible with the Hecke operators, while the map θ r + 1 is Hecke-equivariant up to a twist: we have T ⁢ ( ℓ ) ∘ θ r + 1 = ℓ r + 1 ⁢ θ r + 1 ∘ T ⁢ ( ℓ ) for primes ℓ ∤ N ⁢ p , and similarly for U ⁢ ( ℓ ) with ℓ ∣ N ⁢ p .

Proposition A4.2

Proposition A4.2 ([3, Theorem 1])

Let f β = E r + 2 crit ⁢ ( ψ , τ ) be the critical-slope 𝑝-stabilisation of a 𝑝-decent Eisenstein series E r + 2 ⁢ ( ψ , τ ) , as before. Then, for each sign ± , the eigenspace

H c 1 ⁢ ( Y ̄ , D r ) ⁢ [ T = f β ] ±

where complex conjugation acts by ± 1 is one-dimensional.

This shows that there is an “extra” eigenspace in the kernel of the classical specialisation map ρ r , and complex conjugation acts on this space by − ε ⁢ ( f ) , where ε ⁢ ( f ) = ψ ⁢ ( − 1 ) .

Definition A4.3

We define

V c ( f β ) : = H c 1 ( Y ̄ , D r ) [ T = f β ] ,

which is a 2-dimensional representation of Gal ⁡ ( Q ̄ / Q ) .

In practice, we are interested in a more restrictive situation. For the next result,

M r + 2 † ⁢ ( Γ ) ( T = f β )

stands for the generalised eigenspace of overconvergent modular forms associated to f β .

Definition A4.4

Definition A4.4 ([3, Definition 2.14])

We say f β is non-critical if the generalised eigenspace of overconvergent modular forms M r + 2 † ⁢ ( Γ ) ( T = f β ) associated to f β is 1-dimensional.

Note that here the notion of being critical has to do, as in Bellaïche, with the existence of a generalised eigenspace. In particular, as discussed in [3, §1.3, §2.2.2], critical implies critical-slope, but not conversely. (Observe that, in the notation E r + 2 crit ⁢ ( ψ , τ ) , the superscript “crit” refers to this eigenform being critical-slope, but it is not necessarily critical.)

Theorem A4.5

Theorem A4.5 (Bellaïche–Chenevier; see [5, Remark 1.5])

The following statements are equivalent.

  • The form f β is non-critical.

  • The localisation map

    H f 1 ⁢ ( Q , ψ ⁢ τ − 1 ⁢ ε 1 + r ) → H f 1 ⁢ ( Q p , ψ ⁢ τ − 1 ⁢ ε 1 + r )

    is non-zero.

  • We have L p ⁢ ( ψ − 1 ⁢ τ , r + 1 ) ≠ 0 , where L p denotes the Kubota–Leopoldt 𝑝-adic Dirichlet 𝐿-function.

  • The Galois representation in case (3) of Theorem A2.1 is not locally split at 𝑝.

It is conjectured that these equivalent statements are always true (see [5, Remark 1.5]). The following partial result shows that they are true “often”.

Proposition A4.6

The following statements hold.

  1. For any given 𝜓 and 𝜏, the Eisenstein series E r + 2 crit ⁢ ( ψ , τ ) is non-critical for all but finitely many integers r ⩾ 0 satisfying the parity condition ( − 1 ) r = ψ ⁢ τ ⁢ ( − 1 ) .

  2. If r = 0 , and ψ , τ are such that the Eisenstein series E 2 crit ⁢ ( ψ , τ ) is 𝑝-decent, then this form is also non-critical.

Proof

Part (i) follows from the second of the equivalent conditions of Theorem A4.5: the 𝑝-adic 𝐿-function L p ⁢ ( ψ − 1 ⁢ τ , s ) is not identically 0 on the components of weight space determined by the parity condition, so it cannot vanish for infinitely many integers lying in these components.

For part (ii), we use the fact that L p ⁢ ( χ , 1 ) ≠ 0 for any non-trivial even Dirichlet character 𝜒, which is a consequence of the fact that Leopoldt’s conjecture is known to hold for cyclotomic fields; see e.g. [37, Corollary 5.30]. ∎

We assume henceforth that f β is non-critical.

Theorem A4.7

Theorem A4.7 ([3, Theorem 4])

The following statements hold.

  • Both generalised eigenspaces H c 1 ⁢ ( Y ̄ , D r ) ( T = f β ) ± are one-dimensional over 𝐿.

  • The ε ⁢ ( f ) -eigenspace maps bijectively to its counterpart in H c 1 ⁢ ( Y ̄ , V r ) .

  • The − ε ⁢ ( f ) -eigenspace is isomorphic, via the θ r + 1 map, to H c 1 ⁢ ( Y ̄ , D − 2 − r ) ε ⁢ ( f ) ⁢ [ T = g ] , where g = E − r ord ⁢ ( τ , ψ ) (note the reversal of the order of the characters) is the unique eigenform satisfying θ r + 1 ⁢ ( g ) = f β .

Proposition A4.8

There is a short exact sequence of 𝐿-vector spaces

0 → τ ⁢ ε − 1 − r → V c ⁢ ( f β ) → ψ → 0 .

Proof

We know that the space V c ⁢ ( f β ) has a one-dimensional quotient isomorphic to 𝜓, by (A4.1) and Remark A3.2. On the other hand, also by (A4.1), the kernel is isomorphic to the ( − 1 − r ) -th twist of the H c 1 eigenspace associated to the non-classical ordinary 𝑝-adic Eisenstein series 𝑔 of weight − r . As f β is non-critical, this space must be one-dimensional. So it suffices to show that Gal ⁡ ( Q ̄ / Q ) acts on the one-dimensional space H c 1 ⁢ ( Y ̄ , D − 2 − r ) ⁢ [ T = g ] via 𝜏.

We now recall (see [33, (2.3.10)] for example) that if W − 2 − r denotes the component of weight space containing the integer − 2 − r , there exists a 𝑝-adic family of Eisenstein series over W − 2 − r , whose specialisation in a classical weight k ∈ W − 2 − r ∩ Z ⩾ 0 is the classical ordinary Eisenstein series E k + 2 ord ⁢ ( τ , ψ ) , and whose specialisation at − 2 − r is g = E − r ord ⁢ ( τ , ψ ) .

From the control theorem for étale cohomology in (possibly non-cuspidal) Hida families proved in [33] (which is also valid, with the same proof, for compactly supported cohomology), it follows that there is an open disc 𝑈 around − 2 − r in weight space, and a finite-rank free O ⁢ ( U ) -module with an action of G Q , whose specialisation at a weight κ ∈ U is the eigenspace H c 1 ⁢ ( Y ̄ , D κ ) ⁢ [ T = E κ + 2 ord ⁢ ( τ , ψ ) ] . By applying Remark A3.2 to the specialisation at each integer k ⩾ 0 in 𝑈 (and using the fact that such points are Zariski-dense in 𝑈), we see that this module must be of rank 1 over O ⁢ ( U ) , and G Q acts on it by the character 𝜏. Specialising back to κ = − 2 − r , we obtain the result. ∎

Remark A4.9

This shows that the isomorphism class of V c ⁢ ( f β ) must be one of the isomorphism classes (1) or (2) in Theorem A2.1. We shall see shortly that it is in fact non-split, so it lies in the isomorphism class (2), and is not de Rham at 𝑝.

A4.2 Non-compact supports

We now consider the non-compactly supported space, continuing to assume f β is 𝑝-decent and non-critical.

Definition A4.10

We define

V ⁢ ( f β ) = H 1 ⁢ ( Y ̄ , D r ) ⁢ [ T = f β ] .

There is a canonical map H c 1 ⁢ ( Y ̄ , D r ) → H 1 ⁢ ( Y ̄ , D r ) ; its kernel is the space of boundary modular symbols. This gives a map of Galois representations V c ⁢ ( f β ) → V ⁢ ( f β ) .

Proposition A4.11

Proposition A4.11 ([5, Remark 5.10])

The intersection of V c ⁢ ( f β ) with the boundary symbols is one-dimensional, and is exactly the τ ⁢ ε − 1 − r subrepresentation.

Hence both eigenspaces H 1 ⁢ ( Y , D r ) ⁢ [ T = f β ] ± must have dimension at least 1. In fact, these dimensions are both exactly 1, and are equal to the corresponding generalised eigenspaces, since Bellaiche’s argument in [3, Theorem 3.30] works for non-compactly supported cohomology also. We deduce that there is a short exact sequence of 𝐿-linear Galois representations

(A4.2) 0 → ψ → V ⁢ ( f β ) → τ ⁢ ε − 1 − r → 0 ,

so V c ⁢ ( f β ) and V ⁢ ( f β ) have isomorphic semisimplifications, but the natural map between them is not an isomorphism, but rather identifies the one-dimensional 𝜓-isotypic quotient of V c ⁢ ( f β ) given by Proposition A4.8 with the one-dimensional 𝜓-isotypic subspace of V ⁢ ( f β ) given by (A4.2).

Remark A4.12

As with V c ⁢ ( f β ) above, we have not yet determined whether V ⁢ ( f β ) is a split or non-split extension, so it could be either of the isomorphism classes (1) or (3) of Theorem A2.1, but in either case, it is de Rham at 𝑝. We shall see shortly that it is in fact the non-split extension (3).

A5 Duality and Atkin–Lehner

As in [31], there is a second family of sheaves D r ′ (denoted in [31] by the more verbose notation D r , m ⁢ ( H 0 ′ ) ), which also interpolates the standard finite-dimensional sheaves, but has an action of U ′ ⁢ ( p ) rather than U ⁢ ( p ) . It is the sheaf D r ′ which is used for interpolating Euler system classes in families.

Remark A5.1

There is a canonical 𝐿-linear pairing between D r and D r ′ , landing in the constant sheaf 𝐿, but it is not perfect in general (indeed, we shall see shortly that it does not induce a perfect duality on the fibre at a critical-slope Eisenstein point).

The two sheaves are interchanged by the Atkin–Lehner involution, modulo a twist of the Galois action, so the above structural results for 𝒟 carry over mutatis mutandis to D ′ . So, for r ∈ Z ⩾ 0 , we have exact sequences of sheaves on 𝑌,

0 → D − 2 − r ′ ⁢ ( 1 + r ) → θ r + 1 D r ′ → ρ r V r ∗ → 0 ,

where V r ∗ is the linear dual of V r (isomorphic to the Tate twist V r ⁢ ( r ) ).

Definition A5.2

For f β = E r + 2 crit ⁢ ( ψ , τ ) as above, we define

V ⁢ ( f β ) ∗ = H 1 ⁢ ( Y ̄ , D r ′ ⁢ ( 1 ) ) ⁢ [ T ′ = f β ] ,

which is a 2-dimensional 𝐿-linear representation of Gal ⁡ ( Q ̄ / Q ) fitting into an exact sequence

0 → τ − 1 ⁢ ε 1 + r → V ⁢ ( f β ) ∗ → ψ − 1 → 0 .

We define V c ⁢ ( f β ) ∗ analogously using compactly supported cohomology, so that

0 → ψ − 1 → V c ⁢ ( f β ) ∗ → τ − 1 ⁢ ε 1 + r → 0 ,

and there is a natural map V c ⁢ ( f β ) ∗ → V ⁢ ( f β ) ∗ whose image is the τ − 1 ⁢ ε 1 + r subrepresentation of the latter.

Remark A5.3

We have therefore defined four Galois representations V ⁢ ( f β ) , V ⁢ ( f β ) ∗ , V c ⁢ ( f β ) , and V c ⁢ ( f β ) ∗ associated to f β , all of which are 2-dimensional, with natural 1-dimensional invariant subspaces, and we have identified the Galois actions on their graded pieces.

We shall see in the next section that all four representations are non-split extensions. From this, it follows that V ⁢ ( f β ) ∗ is isomorphic to the dual of V ⁢ ( f β ) (with both being de Rham at 𝑝), while V c ⁢ ( f β ) ∗ is isomorphic to the dual of V c ⁢ ( f β ) (with both being non-de Rham at 𝑝), justifying our choice of notation. However, the pairing giving this duality is not quite the natural Poincaré duality pairing. What we obtain from a naive application of Poincaré duality is pairings

(A5.1) V c ⁢ ( f β ) × V ⁢ ( f β ) ∗ → L , V ⁢ ( f β ) × V c ⁢ ( f β ) ∗ → L

which are not perfect, but rather factor through the 1-dimensional classical quotients of these 2-dimensional representations.

A6 𝑝-adic families

A6.1 Pseudocharacters

The form f β is an overconvergent cuspidal eigenform of finite slope, so it defines a point on the Coleman–Mazur–Buzzard cuspidal eigencurve C 0 of tame level 𝑁. Moreover, our assumptions that 𝑓 be decent and non-critical imply that C 0 is smooth at f β and locally étale over weight space (see [3, Proposition 2.11]), so we may choose a closed disc 𝑈 around f β which maps isomorphically to its image in weight space.

We let 𝐟 be the universal eigenform over 𝑈, which is an overconvergent cuspidal eigenform with coefficients in O ⁢ ( U ) , and weight k + 2 , where k : Z p × → O ⁢ ( U ) × is the canonical character; by definition, the specialisation of 𝐟 at r ∈ U is f β . Let us write β f ∈ O ⁢ ( U ) × for the U ⁢ ( p ) -eigenvalue of 𝐟, so that β f ⁢ ( r ) = β = p r + 1 ⁢ τ ⁢ ( p ) .

There is a canonical Galois pseudocharacter t f : G Q → O ⁢ ( U ) satisfying

t f ⁢ ( Frob ℓ − 1 ) = a ℓ ⁢ ( f )

for good primes ℓ. If 𝔪 is the maximal ideal of O ⁢ ( U ) corresponding to 𝑟, then t f is reducible modulo 𝔪 and its reduction is the sum of two distinct characters. Up to shrinking 𝑈, we have the following result.

Theorem A6.1

Theorem A6.1 ([4, Theorem 1])

The reducibility ideal of the pseudocharacter t f is exactly 𝔪.∎

(More precisely, this is proved in [4] for N = 1 . However, as noted in [3, Remark 2.9], this is purely because a construction of the eigencurve of tame level greater than 1 was not available in the literature at the time [4] was written, and all of the arguments of [4] go over without change to 𝑝-decent Eisenstein series of higher level.)

A6.2 Families of representations

We now define two specific O ⁢ ( U ) -linear Galois representations which both have traces equal to t f , and two more whose traces are the dual t f ∗ . As shown in [31, §4], there exist sheaves D U and D U ′ of O ⁢ ( U ) -modules on 𝑌, whose specialisations at any κ ∈ U are the sheaves D κ and D κ ′ considered above.

Definition A6.2

We define

V ⁢ ( f ) := H ét 1 ⁢ ( Y ̄ , D U ) ⁢ [ T = f ] , V c ⁢ ( f ) := H ét , c 1 ⁢ ( Y ̄ , D U ) ⁢ [ T = f ] , V ⁢ ( f ) ∗ := H ét 1 ⁢ ( Y ̄ , D U ′ ⁢ ( 1 ) ) ⁢ [ T ′ = f ] , V c ⁢ ( f ) ∗ := H ét , c 1 ⁢ ( Y ̄ , D U ′ ⁢ ( 1 ) ) ⁢ [ T ′ = f ]

It follows from the results of [2, 31] that if f β is non-critical, then (up to possibly shrinking 𝑈) the module V ⁢ ( f ) is free of rank 2 over O ⁢ ( U ) and a direct summand of H ét 1 ⁢ ( Y ̄ , D U ) , so we have a natural Hecke-equivariant map

Pr f : H ét 1 ⁢ ( Y ̄ , D U ) → V ⁢ ( f ) ,

and V ⁢ ( f ) carries a O ⁢ ( U ) -linear Galois representation whose trace is t f . Since t f has maximal reducibility ideal, it follows that the fibre of V ⁢ ( f ) at k = r , which is exactly V ⁢ ( f β ) , must be a non-split extension. The same applies mutatis mutandis to the other three modules, showing that all four of V ⁢ ( f β ) , V c ⁢ ( f β ) , V ⁢ ( f β ) ∗ and V c ⁢ ( f β ) ∗ are non-split extensions.

Corollary A6.3

The following statements hold.

  1. The isomorphism class of the representation V ⁢ ( f β ) is the unique non-split, de Rham extension in case (3) of Theorem A2.1. The isomorphism class of V c ⁢ ( f β ) is the non-de Rham extension in case (2) of the theorem.

  2. V ⁢ ( f β ) ∗ is isomorphic to the dual of V ⁢ ( f β ) , and V c ⁢ ( f β ) ∗ to the dual of V c ⁢ ( f β ) .

A6.3 Comparison of O ⁢ ( U ) -lattices

There is a natural “forget supports” map

V c ⁢ ( f ) → V ⁢ ( f ) ,

which is O ⁢ ( U ) -linear. Since this map specialises to an isomorphism at any non-critical-slope classical point, it must be injective, with torsion cokernel; thus we may regard both V c ⁢ ( f ) and V ⁢ ( f ) as O ⁢ ( U ) -lattices in V ⁢ ( f ) ⊗ O ⁢ ( U ) Frac ⁡ O ⁢ ( U ) . Shrinking our disc 𝑈 further if necessary, we may assume that the cokernel is supported at 𝔪.

Since the map V c ⁢ ( f β ) → V ⁢ ( f β ) is not the zero map, V c ⁢ ( f ) is not contained in m ⋅ V ⁢ ( f ) . Thus we may find a basis ( e 1 , e 2 ) of V ⁢ ( f ) , and an integer r ⩾ 1 , such that ( e 1 , X r ⁢ e 2 ) is a basis of V c ⁢ ( f ) , where 𝑋 is a uniformiser of 𝔪. From Theorem A6.1, we must in fact have r = 1 . Thus X ⁢ V ⁢ ( f ) ⊂ V c ⁢ ( f ) , and we have a chain of inclusions

⋯ ⊃ 1 X ⁢ V c ⁢ ( f ) ⊃ V ⁢ ( f ) ⊃ V c ⁢ ( f ) ⊃ X ⁢ V ⁢ ( f ) ⊃ ⋯

with all of the successive quotients l-dimensional over 𝐿, and alternately equal to either 𝜓 or τ ⁢ ε − 1 − r as Galois modules. Similarly, we have a chain

(A6.1) ⋯ ⊃ 1 X ⁢ V c ⁢ ( f ) ∗ ⊃ V ⁢ ( f ) ∗ ⊃ V c ⁢ ( f ) ∗ ⊃ X ⁢ V ⁢ ( f ) ∗ ⊃ ⋯

with quotients alternately isomorphic to either ψ − 1 or τ − 1 ⁢ ε 1 + r .

A6.4 Duality

We recall the construction of the “Ohta pairing” from [31, §4.3], which is a O ⁢ ( U ) -bilinear pairing on D U × D U ′ , taking values in the constant sheaf O U . This gives rise to a pairing of Galois representations { − , − } : V c ⁢ ( f ) × V ⁢ ( f ) ∗ → O ⁢ ( U ) , interpolating the Poincaré duality pairings on the classical specialisations.

This pairing is not perfect, since modulo 𝔪 it becomes the pairing

V c ⁢ ( f β ) × V ⁢ ( f β ) ∗ → L

which we have seen in (A5.1) is non-perfect. However, since the Poincaré duality pairings on non-critical-slope, cuspidal specialisations of 𝐟 are perfect, the map

V c ⁢ ( f ) → Hom O ⁢ ( U ) ⁡ ( V ⁢ ( f ) ∗ , O ⁢ ( U ) )

induced by the Ohta pairing must be injective, with torsion cokernel. Shrinking 𝑈 if needed, we may suppose the cokernel is supported at 𝔪.

Proposition A6.4

The O ⁢ ( U ) -submodule

{ x ∈ V c ⁢ ( f ) ⁢ [ 1 / X ] : { x , y } ∈ O ⁢ ( U ) ⁢ for all ⁢ y ∈ V ⁢ ( f ) ∗ }

is equal to V ⁢ ( f ) ⊂ 1 X ⁢ V c ⁢ ( f ) , and hence the Ohta pairing extends uniquely to a perfect O ⁢ ( U ) -linear pairing V ⁢ ( f ) × V ⁢ ( f ) ∗ → O ⁢ ( U ) .

Proof

Let us write 𝑊 for the O ⁢ ( U ) -lattice

{ x ∈ V c ⁢ ( f ) ⁢ [ 1 / X ] : { x , y } ∈ O ⁢ ( U ) ⁢ for all ⁢ y ∈ V ⁢ ( f ) ∗ } .

Evidently, W ⊇ V c ⁢ ( f ) , and the quotient is 𝑋-torsion. The image of V c ⁢ ( f ) / X ⁢ V c ⁢ ( f ) ≅ V c ⁢ ( f β ) in W / X ⁢ W ≅ Hom L ⁡ ( V ⁢ ( f β ) ∗ , L ) is not zero: it is exactly the 1-dimensional classical quotient of V c ⁢ ( f β ) . So the quotient W / V c ⁢ ( f ) is isomorphic to O ⁢ ( U ) / X n , for some 𝑛.

However, since both 𝑊 and V c ⁢ ( f β ) are G Q -invariant O ⁢ ( U ) -lattices in V c ⁢ ( f ) ⁢ [ 1 / X ] , this implies that the action of Galois in a suitable O ⁢ ( U ) -basis of 𝑊 is upper-triangular modulo X n , and if n ⩾ 2 , this contradicts the maximality of the reducibility ideal of the pseudocharacter t f .

Hence 𝑊 is a Galois-invariant sublattice of 1 X ⁢ V c ⁢ ( f ) containing V c ⁢ ( f ) , with both containments strict. As V c ⁢ ( f ) / X ≅ V c ⁢ ( f β ) has a unique Galois-invariant line, there is a unique such lattice, namely V ⁢ ( f ) . ∎

Remark A6.5

In particular, there is a uniquely determined perfect pairing

V ⁢ ( f β ) × V ⁢ ( f β ) ∗ → L ,

refining the result above that these representations are abstractly dual to each other.

A7 Nearly overconvergent cohomology

We summarise here some results from [31, §5.2] on “nearly overconvergent étale cohomology”. The basic object of study is the cohomology of the sheaves D U − j ′ ⊗ V j , for j ∈ Z ⩾ 0 , which we interpret as “nearly overconvergent cohomology of degree 𝑗”. Here U − j denotes the image of the disc 𝑈 under the map W → W , κ ↦ κ − j , so that, for every integer k ∈ U ∩ Z with k ⩾ j , the fibre of D U − j ′ ⊗ V j at 𝑘 surjects onto V k − j ⊗ V j .

Remark A7.1

These modules are relevant to our present study because the Beilinson–Flach classes associated to Coleman families constructed in [31] naturally land in these larger sheaves, rather than in the D U ′ sheaves themselves, as we shall recall in more detail below.

As explained [31, §5.2], there is a natural map of sheaves β j ∗ : D U ′ → D U − j ′ ⊗ V j , compatible with the Clebsch–Gordan maps V k ↪ V k − j ⊗ V j for integers k ⩾ j , and there is a map the other way, δ j ∗ : D U − j ′ ⊗ V j → D U ′ , such that δ j ∗ ∘ β j ∗ is multiplication by ( ∇ j ) ∈ O U . Both of these maps are O ⁢ ( U ) -linear, and compatible with Hecke correspondences away from 𝑝, and with U ′ ⁢ ( p ) .

Since O ⁢ ( U ) is reduced and ( ∇ j ) ≠ 0 , we may make sense of the map

Pr : [ j ] = 1 ( ∇ j ) δ j ∗ : H 1 ( Y ̄ , D U − j ′ ⊗ V j ) → 1 ( ∇ j ) H 1 ( Y ̄ , D U ′ ) ,

whose specialisation at any k ⩾ j is a left inverse of β j ∗ . The denominator has simple poles at all locally algebraic characters of degree k ∈ { 0 , … , ( j − 1 ) } , but the residues at these poles are valued in the kernel of specialisation on D k ′ , since the composite

D k − j ′ ⊗ V j → δ j ∗ D k ′ → V k

is zero.

We shall need this map in the case when 𝐟 is a Coleman family with one critical-slope Eisenstein fibre in weight 𝑟, and j = r + 1 . Shrinking 𝑈 if necessary, we can assume that the specialisations at all points of 𝑈 which are locally algebraic of degree 0 , … , r are cuspidal and non-critical-slope, except possibly at 𝑟 itself. Thus Pr f [ r + 1 ] on cohomology takes values in 1 X ⁢ V ⁢ ( f ) ∗ , but its residue maps trivially into V ⁢ ( f β ) quo ∗ , so in fact, it factors through the slightly smaller module 1 X ⁢ V c ⁢ ( f ) ∗ ⊃ V ⁢ ( f ) ∗ , where 𝑋 is a uniformiser at r ∈ U (cf. equation (A6.1)). Thus we obtain a map of Galois representations over O ⁢ ( U ) ,

Pr f [ r + 1 ] : H 1 ⁢ ( Y ̄ , D U − ( r + 1 ) ′ ⊗ V r + 1 ⁢ ( 1 ) ) → 1 X ⁢ V c ⁢ ( f ∗ ) ,

whose composite with β j ∗ is the natural projection map

Pr f : H 1 ⁢ ( Y ̄ , D U ′ ⁢ ( 1 ) ) → V ⁢ ( f ) ∗ ⊂ 1 X ⁢ V c ⁢ ( f ) ∗ .

A8 𝑝-adic Hodge theory

We now investigate the restriction of the Galois modules constructed above to the decomposition group at 𝑝. We fix an embedding Q ̄ ↪ Q ̄ p , so we may regard Gal ⁡ ( Q ̄ p / Q p ) as a subgroup of Gal ⁡ ( Q ̄ / Q ) .

A8.1 Notation

  • Let ℛ be the Robba ring over Q p , and let R U = R ⊗ ̂ O U .

  • Let t ∈ R be the period for the cyclotomic character, so φ ⁢ ( t ) = p ⁢ t and γ ⁢ ( t ) = ε ⁢ ( γ ) ⁢ t .

  • For δ ∈ O ⁢ ( U ) × , let R U ⁢ ( δ ) be the rank 1 ( φ , Γ ) -module over R U generated by an element 𝑒 which is Γ-invariant and satisfies φ ⁢ ( e ) = δ ⁢ e .

  • For 𝐷 a ( φ , Γ ) -module over ℛ (or R U ), write D cris ⁢ ( D ) = D ⁢ [ 1 / t ] Γ , equipped with the filtration given by Fil n ⁡ D cris ⁢ ( D ) = ( t n ⁢ D ) Γ ; this is compatible with the usual notation for D = D cris ⁢ ( V ) , 𝑉 crystalline.

  • Let D ⁢ ( f ) ∗ = D rig † ⁢ ( V ⁢ ( f ) ∗ ) , and similarly D c ⁢ ( f ) ∗ .

A8.2 Triangulations for D ⁢ ( f ) ∗

We know that the space of homomorphisms

R U ⁢ ( β f ψ ⁢ ( p ) ⁢ τ ⁢ ( p ) ) ⁢ ( 1 + k ) → D ⁢ ( f ) ∗

is a finitely generated O ⁢ ( U ) -module, by the main theorem of [23]. Thus it must be free of rank 1, since it is clearly torsion-free, and there is a Zariski-dense set of x ∈ O ⁢ ( U ) where the fibre is 1-dimensional. Let F + ⁢ D ⁢ ( f ) ∗ be the image of a generator of this map, and F − the quotient, so that we have a short exact sequence 0 → F + ⁢ D ⁢ ( f ) ∗ → D ⁢ ( f ) ∗ → F − ⁢ D ⁢ ( f ) ∗ → 0 .

Over the punctured disc U − { r } , the sub and quotient are both free of rank 1 and so the above sequence defines a triangulation of D ⁢ ( f ) ∗ .

Proposition A8.1

If f β is non-critical, the submodule F + ⁢ D ⁢ ( f β ) ∗ is saturated; thus F − ⁢ D ⁢ ( f ) ∗ is free of rank 1 over R U , and F ± ⁢ D ⁢ ( f ) ∗ define a triangulation of D ⁢ ( f ) ∗ over 𝑈.

Proof

This follows by a comparison of filtration degrees on D cris : if the submodule were not saturated, then the φ = ψ ⁢ ( p ) − 1 -eigenspace in D cris ⁢ ( V ⁢ ( f β ) ∗ ) would be contained in Fil n for some n > − 1 − r , and hence necessarily in Fil 0 . This would force V ⁢ ( f β ) ∗ to locally split at 𝑝 as a direct sum, which according to Theorem A4.5 contradicts the assumption that f β is non-critical. ∎

We thus have two exact sequences “cutting across each other”, one arising from the triangulation (the horizontal one in the diagram), and one from the global reducibility of V ⁢ ( f β ) ∗ (the vertical one):

There is an isomorphism D † ⁢ ( V ⁢ ( f β ) quo ∗ ) ≅ R ⁢ ( ψ ⁢ ( p ) − 1 ) , and the lower left dotted arrow must, therefore, identify its source with the ( φ , Γ ) -stable submodule t r + 1 ⁢ R ⁢ ( ψ ⁢ ( p ) − 1 ) of the target, and similarly for the upper right dotted arrow. This corresponds to the fact that the map

D cris ⁢ ( F + ⁢ D ⁢ ( f β ) ∗ ) → D cris ⁢ ( V ⁢ ( f β ) quo ∗ )

is an isomorphism on the underlying 𝜑-modules, but shifts the filtration degree by r + 1 .

Triangulations for D c ⁢ ( f ) ∗

For D c ⁢ ( f ) ∗ , the situation is a little different: in this case, the triangulation becomes singular in the fibre at 𝑟. More precisely, we may choose generators of the free rank 1 O ⁢ ( U ) -modules

Hom ( φ , Γ ) ⁡ ( R U ⁢ ( β f / ψ ⁢ τ ⁢ ( p ) ) ⁢ ( 1 + k ) , D c ⁢ ( f ) ∗ ) and Hom ( φ , Γ ) ⁡ ( D c ⁢ ( f ) ∗ , R U ⁢ ( β f − 1 ) ) .

Then we obtain a submodule F + ⁢ D c ⁢ ( f ) ∗ and a quotient F − ⁢ D c ⁢ ( f ) ∗ which restrict to the triangulation away from weight 𝑟, and such that the maps

F + ⁢ D c ⁢ ( f β ) ∗ → D c ⁢ ( f β ) ∗ and D c ⁢ ( f β ) ∗ → F − ⁢ D c ⁢ ( f β ) ∗

are nonzero, where F ± ⁢ D c ⁢ ( f β ) ∗ are the fibres of F ± ⁢ D c ⁢ ( f ) ∗ at 𝑟. If we identify V c ⁢ ( f ) ∗ with a submodule of V ⁢ ( f ) ∗ , then we deduce that

F + ⁢ D c ⁢ ( f ) ∗ = X ⋅ F + ⁢ D ⁢ ( f ) ∗ , F − ⁢ D c ⁢ ( f ) ∗ = F − ⁢ D ⁢ ( f ) ∗ ,

where 𝑋 is a uniformiser at r ∈ U .

However, these two maps do not define a triangulation, as the submodule F + ⁢ D c ⁢ ( f β ) ∗ is not saturated: its image in D c ⁢ ( f β ) ∗ is exactly t r + 1 ⋅ D † ⁢ ( V c ⁢ ( f β ) sub ∗ ) . Similarly (and in fact dually), the quotient map D c ⁢ ( f β ) ∗ → F − ⁢ D c ⁢ ( f β ) ∗ has image t r + 1 ⁢ F − ⁢ D c ⁢ ( f β ) ∗ , which we can identify with D † ⁢ ( V c ⁢ ( f β ) quo ∗ ) ≅ R ⁢ ( p r + 1 / β ) ⁢ ( r + 1 ) . So we have an analogous “cross” as before, but the horizontal row is not exact:

The lower right arrow induces an isomorphism after inverting 𝑡, and hence is an isomorphism between D cris modules (since D cris ⁢ ( D ) = D ⁢ [ 1 / t ] Γ ), but since the filtration on D cris is given by ( t n ⁢ D ) Γ , the isomorphism shifts the filtration degrees – the filtration on D cris ⁢ ( V c ⁢ ( f β ) quo ∗ ) is concentrated in degree − 1 − r , but the filtration on D cris ⁢ ( F − ⁢ D c ⁢ ( f β ) ∗ ) is concentrated in degree 0.

A8.3 Crystalline periods

Essentially by construction, we may choose (non-canonically) an isomorphism

b f + : D cris ⁢ ( F + ⁢ D ⁢ ( f ) ∗ ⁢ ( − 1 − k ) ) ≅ O U .

If 𝑔 is a non-critical-slope, cuspidal classical specialisation of 𝐟, with 𝑔 of weight k + 2 for some k ≠ r , then we have a canonical isomorphism between the fibres of the above modules at 𝑘, given by the image modulo F − of the class in Fil k + 1 ⁡ D cris ⁢ ( V ⁢ ( g ) ) of the differential form ω g associated to 𝑔. Here we use the comparison isomorphism between de Rham and étale cohomology crucially.

So, for each such 𝑔, there must exist a non-zero constant c g ∈ L × such that b f + specialises to c g ⁢ ω g .

At X = 0 , we have an isomorphism

D cris ⁢ ( F + ⁢ D ⁢ ( f β ) ∗ ) ≅ D cris ⁢ ( V ⁢ ( f β ) quo ∗ ) ,

and we have a duality between V ⁢ ( f β ) quo ∗ and V c ⁢ ( f β ) quo , so we should regard b f β + as a basis of the space

D cris ⁢ ( V c ⁢ ( f β ) quo ) ≅ H dR , c 1 ⁢ ( Y , V r ) ⁢ [ T = f β ] = H 1 ⁢ ( X , ω − r ⁢ ( − cusps ) ) ⁢ [ T = f β ] .

So there exists some scalar c f β ∈ L × such that b f β + = c f β ⁢ η f β , where η f is as defined in Section A3.2.

Remark A8.2

It is curious to note that the element b f + “generically” interpolates the Fil 1 vectors ω g for specialisations 𝑔 of weight ≠ r , but at the bad weight k = r , it interpolates the Fil 0 vector η f β instead.

B Euler systems and 𝑝-adic 𝐿-functions: Background

In the next few sections, we recall the Euler systems and 𝑝-adic 𝐿-functions we shall use below. We present no new results here, but we will need to re-formulate some well-known results in minor ways, in order to be able to compare different constructions under our “Eisenstein degeneration” formalism in the final sections of this article.

Assumption B1.1

Throughout Part B of this paper, we shall use f β to denote a classical cuspidal𝑝-stabilised newform, of weight r 1 + 2 for some integer r 1 ⩾ 0 , and non-critical slope, and we shall write 𝐟 for a Coleman family of eigenforms, over some open affinoid V 1 ∋ r in weight space, specialising to f β in weight r 1 . We suppose, for convenience, that f β is a 𝑝-stabilisation of an eigenform of prime-to-𝑝 level whose Hecke polynomial at 𝑝 has distinct roots (so f β is a “noble eigenform” in the sense of [18]).

Similarly, g , h will denote families over some affinoids V 2 , V 3 passing through some given noble eigenforms g β , h β of weights r 1 , r 2 . We shall allow ourselves to shrink the discs V i if needed, replacing them with arbitrarily small neighbourhoods of the r i ; this allows us to assume, for instance, that all classical-weight specialisations of our families are noble cuspidal eigenforms.

(Of course, the primary goal of this paper is precisely to study those families which do not satisfy this condition, and this will be the goal of Part C, but firstly, we shall give a systematic account of the theory in the above setting, before explaining the modifications necessary for the critical-slope Eisenstein cases.)

B1.1 Setup

We fix an embedding Q ̄ ↪ Q ̄ p . As in Part A, we shall fix a finite extension L / Q p with integers 𝒪. Let H Γ be the distribution algebra of Γ ≅ Z p × (with 𝐿-coefficients), and j : Γ ↪ H Γ × the universal character, which we regard as a character of the Galois group by composition with the cyclotomic character.

Notation

Abusively, we shall write H 1 ⁢ ( Q , − ) for H 1 ⁢ ( G Q , S , − ) where 𝑆 is a sufficiently large finite set of primes (i.e. containing 𝑝 and all primes at which the relevant representation is ramified).

For any 𝐿-linear G Q -representation 𝑉, we write V ⁢ ( − j ) as a shorthand for V ⊗ L H Γ ⁢ ( − j ) . Thus, for any 𝒪-lattice T ⊂ V , we have a canonical isomorphism of H Γ -modules

H 1 ⁢ ( Q , V ⁢ ( − j ) ) ≅ H Γ ⊗ Λ Γ lim ← n ⁡ H 1 ⁢ ( Q ⁢ ( μ p ∞ ) , T ) ,

and similarly for G Q p -representations.

B1.2 Local machinery: Coleman–Perrin-Riou maps

Let 𝑉 be a crystalline 𝐿-linear representation of G Q p . Then there is a homomorphism of H Γ -modules, the Perrin-Riou regulator,

L V Γ : H 1 ⁢ ( Q p , V ⁢ ( − j ) ) → I − 1 ⁢ H Γ ⊗ L D cris ⁢ ( Q p , V )

(which depends on a choice of 𝑝-power roots of unity ( ζ p n ) n ⩾ 1 in L ̄ , although we suppress this from the notation); here 𝐼 is a certain fractional ideal depending on the Hodge–Tate weights of 𝑉. The map L V Γ is characterised by a compatibility with the Bloch–Kato logarithm and dual-exponential maps for twists of 𝑉. See [28] for explicit formulae.

If we choose a vector η ∈ D cris ⁢ ( V ) , and apply the above constructions to V ∗ , then we obtain a Coleman map

Col η : H 1 ⁢ ( Q p , V ∗ ⁢ ( − j ) ) → I − 1 ⁢ H ⁢ ( Γ ) , x ↦ ⟨ L V ∗ Γ ⁢ ( x ) , η ⟩ .

We shall most often use this when 𝑉 is unramified and non-trivial, in which case 𝐼 is the unit ideal.

All three objects above – the cohomology of V ⁢ ( − j ) , the D cris module, and the regulator map connecting them – can also be defined more generally for crystalline ( φ , Γ ) -modules over the Robba ring, whether or not they are étale.

B2 The GL 2 / Q setting

The Kato Euler system of [21] can be attached to a Coleman family 𝐟 satisfying Assumption B1.1, as discussed e.g. in [18]. However, for our purposes, it suffices to restrict to the case of Hida families, following the developments of [32]. We suppose our family 𝐟 is new of some level 𝑁, and, as in [32], we assume that the Galois representation attached to 𝐟 is residually irreducible.

B2.1 Periods

Let f k be (the newform associated to) the weight k + 2 specialisation of 𝐟, for some k ⩾ 0 . For each sign ± , the eigenspace in Betti cohomology of Y 1 ⁢ ( N ) (with Q ⁢ ( f ) -coefficients) on which the Hecke operators act by the eigensystem of f k is one-dimensional, and we choose bases γ f ± of these spaces. These determine complex periods Ω f k ± ∈ C × .

Let V 1 be the open of the weight space over which the family is defined. Then we have an O ⁢ ( V 1 ) -module V ⁢ ( f ) ∗ interpolating the Betti (or étale) cohomology eigenspaces of all specialisations of 𝐟. Up to possibly shrinking V 1 , we may assume that the eigenspaces V ⁢ ( f ) ( c = ± 1 ) are free of rank 1 over O ⁢ ( V 1 ) , and choose bases γ f ± . In general, we cannot arrange that the weight 𝑘 specialisation of γ f ± is defined over Q ⁢ ( f k ) for all 𝑘, so it is convenient to extend the definition of Ω f k ± accordingly, so these periods now lie in the space ( L ⊗ Q ⁢ ( f k ) C ) × .

Remark B2.1

For avoidance of confusion, we point out that the period we are calling Ω f k ± corresponds to

1 λ ± ⁢ ( k ) ⁢ Ω f k ±

in the notation of [7, §3.2]. Hence, although the periods we have considered are elements in ( L ⊗ Q ⁢ ( f k ) C ) × (which will ease our subsequent discussions), it must be clear that one can easily recover more familiar objects from them.

B2.2 The Kitagawa–Mazur 𝑝-adic 𝐿-function

Having chosen γ f ± , we can define Kitagawa–Mazur-type 𝑝-adic 𝐿-functions (see [26])

L p ⁢ ( f ) ∈ O ⁢ ( V 1 × W ) ,

which interpolate the critical 𝐿-values of all classical, weight at least 2 specialisations of 𝐟, with the periods determined by γ f ± . More precisely, the value at ( k , j ) , with 0 ⩽ j ⩽ k , is given by

L p ⁢ ( f ) ⁢ ( k , j ) = j ! ⁢ ( 1 − β k p 1 + j ) ⁢ ( 1 − p j α k ) ( − 2 ⁢ π ⁢ i ) j ⁢ Ω f k ± ⁢ L ⁢ ( f k , 1 + j ) ,

where f k is the weight k + 2 specialisation of 𝐟 as above, ( α k , β k ) are the roots of its Hecke polynomial (with α k corresponding to the family 𝐟), and ± = ( − 1 ) j . (Here we assume f k is new of level 𝑁, which is automatic for k > 0 ; a slightly modified formula applies if k = 0 and f k is a newform of level p ⁢ N and Steinberg type at 𝑝.)

B2.3 The adjoint 𝑝-adic 𝐿-function

We shall also need the following construction. For any classical specialisation 𝑓 of 𝐟, if we define the plus and minus periods Ω f ± using Q ⁢ ( f ) -rational basis vectors γ f ± , then the ratio

L alg ( Ad f , 1 ) : = − 2 k − 1 ⁢ i ⁢ π 2 ⁢ ⟨ f , f ⟩ Ω f + ⁢ Ω f −

is in Q ⁢ ( f ) × . If we choose bases γ f ± over the family as above, and use the periods for each f k determined by these, then a construction due to Hida [19] gives a 𝑝-adic adjoint 𝐿-function L p ⁢ ( Ad ⁡ f ) ∈ O ⁢ ( V 1 ) interpolating these ratios,

L p ⁢ ( Ad ⁡ f ) ⁢ ( k ) = ( 1 − β k α k ) ⁢ ( 1 − β k p ⁢ α k ) ⁢ L alg ⁢ ( Ad ⁡ f k , 1 )

for all k ∈ V 1 ∩ Z ⩾ 0 such that f k is a level 𝑁 newform. In particular, recall that if 𝐟 is 𝑝-distinguished, then the congruence ideal of 𝐟 is principal, and this ideal is generated by L p ⁢ ( Ad ⁡ f ) .

B2.4 Kato’s Euler system

Having chosen γ f ± , we obtain a canonical Kato class

κ ⁢ ( f ) ∈ H 1 ⁢ ( Q , V ⁢ ( f ) ∗ ⁢ ( − j ) ) ,

which is the “𝑝-direction” of an Euler system. Kato’s explicit reciprocity law [21] establishes that the image of that class under the Perrin-Riou map recovers the 𝑝-adic 𝐿-function.

More precisely, let F + ⁢ V ⁢ ( f ) denote the rank 1 unramified subrepresentation of V ⁢ ( f ) over O ⁢ ( V 1 ) (which exists since 𝐟 is ordinary), and let η f ∈ D cris ⁢ ( F + ⁢ V ⁢ ( f ) ) be the canonical vector constructed in [24], which is characterised by interpolating the classes η f of Section A3.2 for each classical specialisation 𝑓 of 𝐟. Then we have

⟨ L F − ⁢ V ⁢ ( f ) ∗ Γ ⁢ ( loc p ⁡ κ ⁢ ( f ) ) , η f ⟩ = L p ⁢ ( f ) .

Remark B2.2

This is a slight strengthening of a result of Ochiai; see [32, Theorem 3.17] for the original formulation. Ochiai’s result is a little less precise, since he chooses an arbitrary basis 𝑑 of the module D cris ⁢ ( F + ⁢ V ⁢ ( f ) ) (which is denoted 𝒟 in [32]); we have used the results on Eichler–Shimura in families proved in [24] to choose a canonical basis η f , for which the correction terms C p , p , d in Ochiai’s formulae are all 1.

B3 Double and triple products

B3.1 The Rankin–Selberg setting

Now let 𝐟 and 𝐠 be two Coleman families satisfying Assumption B1.1, living over discs V 1 , V 2 in weight space.

𝑝-adic 𝐿-functions

There is an “𝐟-dominant” 𝑝-adic 𝐿-function L p f ⁢ ( f , g ) over V 1 × V 2 × W , whose value at ( k , ℓ , j ) with ℓ + 1 ⩽ j ⩽ k is

( ⋆ ) ⋅ L ⁢ ( f k , g ℓ , 1 + j ) ⟨ f k , f k ⟩ ,

where ( ⋆ ) is the usual mélange of Euler factors, powers of 𝑖 and 𝜋, etc. Similarly, there is a “𝐠-dominant” 𝑝-adic 𝐿-function L p g ⁢ ( g , f ) , with an interpolating range at points with k + 1 ⩽ j ⩽ ℓ . If 𝐟 is ordinary (i.e. a Hida family), then L p f ⁢ ( f , g ) is bounded.

Remark B3.1

Note that our 𝑝-adic 𝐿-functions L ⁢ ( f , g , s ) here are imprimitive, i.e. their Dirichlet coefficients are given by the usual straightforward formula in terms of 𝑞-expansion coefficients; cf. [24, §2.7]. This means they may differ by finitely many Euler factors from the primitive𝑝-adic 𝐿-function associated to the Galois representation V p ⁢ ( f ) ⊗ V p ⁢ ( g ) (although this issue only arises if the levels of 𝑓 and 𝑔 are not coprime).

Euler systems

The Beilinson–Flach Euler system of [31] is attached to two modular forms, or more generally to two Coleman families 𝐟 and 𝐠. This generalises the earlier construction of [24], where the variation was restricted to the case of ordinary families. Consider the rank 4 module

V ⁢ ( f , g ) ∗ ⁢ ( − j ) over O ⁢ ( V 1 × V 2 × W ) ,

characterised by the property that, on specialising at any integers ( k , ℓ , j ) , with k ∈ V 1 and ℓ ∈ V 2 , we recover V ⁢ ( f k ) ∗ ⊗ V ⁢ ( g ℓ ) ∗ ⁢ ( − j ) .

Fix d ∈ Z > 1 such that ( d , 6 ⁢ S ) = 1 . There exists a cohomology class of Beilinson–Flach elements

κ d ⁢ ( f , g ) ∈ H 1 ⁢ ( Q , V ⁢ ( f , g ) ∗ ⁢ ( − j ) ) ,

which is the one in [31, Theorem 5.4.2], where it would be denoted by B d ⁢ F 1 , 1 [ f , g ] . (Note that we have slightly modified the notation just for being consistent with the other Euler systems, and we have written 𝑑 and not 𝑐 for the auxiliary parameter to avoid any misunderstanding with the index 𝑐 used to denote compactly supported cohomology.)

The dependence on 𝑑 is as follows: after tensoring with Frac ⁡ O ⁢ ( V 1 × V 2 × W ) , the class

κ ( f , g ) : = C d − 1 ⊗ κ d ( f , g )

is independent of 𝑑, where

C d ( f , g , j ) : = d 2 − d ( j − k 1 − k 2 ) ε f − 1 ( d ) ε g − 1 ( d ) .

Reciprocity laws

With the notation used in [31, §6], let D ⁢ ( f ) ∗ be the ( φ , Γ ) -module of V ⁢ ( f ) ∗ , and consider the rank 1 submodule F + ⁢ D ⁢ ( f ) ∗ ⊂ D ⁢ ( f ) ∗ together with the corresponding quotient F − ⁢ D ⁢ ( f ) ∗ . We consider the same filtration for D ⁢ ( g ) ∗ . We write

F − − ⁢ D ⁢ ( f , g ) ∗ = F − ⁢ D ⁢ ( f ) ∗ ⊗ ̂ F − ⁢ D ⁢ ( g ) ∗ ,

and similarly for F − + , F + − , F + + . We also define F − ∘ ⁢ D ⁢ ( f , g ) ∗ = F − ⁢ D ⁢ ( f ) ∗ ⊗ ̂ D ⁢ ( g ) ∗ . Proceeding as in [31, Theorem 7.1.2], the projection of κ d ⁢ ( f , g ) to F − − ⁢ D ⁢ ( f , g ) ∗ vanishes. Hence the projection to F − ∘ is in the image of the injection

H 1 ⁢ ( Q , F − + ⁢ D ⁢ ( f , g ) ∗ ⁢ ( − j ) ) → H 1 ⁢ ( Q , F − ∘ ⁢ D ⁢ ( f , g ) ∗ ⁢ ( − j ) ) .

Then the reciprocity law of Loeffler–Zerbes [31, Theorem 7.1.5] establishes that the image of that class under the Perrin-Riou map recovers the 𝑝-adic 𝐿-function,

Col η f ⊗ ω g ⁡ ( loc p ⁡ ( κ d ⁢ ( f , g ) ) ) = C d ⁢ ( f , g , j ) ⋅ L p f ⁢ ( f , g ) ,

where the Coleman map is the composition of the Perrin-Riou regulator followed by the pairing with the appropriate differentials.

B3.2 Diagonal cycles

Let ( f , g , h ) be a triple of Coleman families, living over discs V 1 , V 2 , V 3 in weight space, with all classical specialisations non-critical-slope cusp forms, as in the previous section.

We suppose the tame nebentype characters satisfy ε f ⁢ ε g ⁢ ε h = 1 . It follows that we may choose (non-uniquely) a character t : Z p × → O ⁢ ( V 1 × V 2 × V 3 ) satisfying 2 ⁢ t = k 1 + k 2 + k 3 , where k i are the universal characters into O ⁢ ( V i ) . This imposes an additional condition on our specialisations: we shall say a point 𝑃 of V 1 × V 2 × V 3 is an “integer point” if ( k 1 , k 2 , k 3 ) specialise at 𝑃 to integers ( k , ℓ , m ) ⩾ − 1 and, in addition, 𝐭 specialises to k + ℓ + m 2 (rather than its product with the quadratic character mod 𝑝), which amounts to requiring that k + ℓ + m lie in a particular congruence class modulo 2 ⁢ ( p − 1 ) .

𝑝-adic 𝐿-functions

Following a construction of Andreatta and Iovita [1], there is a 𝐟-dominant square root 𝑝-adic 𝐿-function L p f ⁢ ( f , g , h ) over V 1 × V 2 × V 3 . The square of its value at an integer point ( k , ℓ , m ) with k > ℓ + m is

( ⋆ ) ⋅ L ⁢ ( f k , g ℓ , h m , k + ℓ + m 2 + 2 ) ,

where ( ⋆ ) is the usual mélange of Euler factors, complex periods, etc. (Note that, with our conventions, f k has weight k + 2 , etc., so k + ℓ + m 2 + 2 is the centre of the functional equation.)

Remark B3.2

In fact, we shall only need this construction when the “dominant” family is an ordinary family. In this case, the construction is actually considerably simpler, and can be carried out via the same techniques as in the ordinary case, without need to resort to the developments of Andreatta–Iovita.

Euler systems

Consider the rank 8 module

(B3.1) V ⁢ ( f , g , h ) ∗ := V ⁢ ( f ) ∗ ⊗ V ⁢ ( g ) ∗ ⊗ V ⁢ ( h ) ∗ ⁢ ( − 1 − k 1 + k 2 + k 3 2 ) over ⁢ O ⁢ ( V 1 × V 2 × V 3 ) ,

which is Tate self-dual. By the works of Darmon–Rotger [17] and Bertolini–Seveso–Venerucci [11], there is a diagonal-cycle class κ ⁢ ( f , g , h ) attached to the triple ( f , g , h ) (and the choice of square-root character 𝐭, which we suppress); this is a class

κ ⁢ ( f , g , h ) ∈ H 1 ⁢ ( Q , V ⁢ ( f , g , h ) ∗ ) ,

introduced for instance in [11, §8.1]. It is characterised by the property that, on specialising at any integer point ( k , ℓ , m ) , with k ∈ V 1 , ℓ ∈ V 2 and m ∈ V 3 satisfying the “balanced” conditions { k ⩽ m + ℓ , ℓ ⩽ m + k , m ⩽ k + ℓ } , we recover the Abel–Jacobi image of the diagonal cycle for ( f k , g ℓ , h m ) defined in [16].

Remark B3.3

Note that there is an omission in [11, Section 4.2], where the machinery for interpolating diagonal-cycle classes is developed: it is implicitly assumed that Ohta’s pairing V ⁢ ( f ) × V c ⁢ ( f ) ∗ → O ⁢ ( V 1 ) (see [11, formula (82)]) is perfect. This cannot be true in general since, at a critical-slope Eisenstein point, it specialises to the second of the two pairings in (A5.1) above, and we have seen that this has 1-dimensional left and right kernels. However, the pairing does become perfect after specialising to a neighbourhood of a noble eigenform, and hence the construction is valid when f , g , h satisfy the conditions of Assumption B1.1; see the erratum [12] for the necessary modifications.

We shall see in Part C of this paper that the construction gives something slightly weaker, i.e. a cohomology class taking values in 1 X ⁢ V c ⁢ ( f , g , h ) ∗ ⊃ V ⁢ ( f , g , h ) ∗ , when one of the families is critical-slope.

Reciprocity laws

With the notation of Section B3.1 above, we may consider the ( φ , Γ ) -module D ⁢ ( f , g , h ) ∗ , as well as its different filtrations. Adapting [11, Corollary 8.2] to our setting yields that the diagonal cycle class lies in the rank four submodule ( F + ⁣ + ∘ + F + ⁣ ∘ + + F ∘ ⁣ + + ) ⁢ D ⁢ ( f , g , h ) ∗ . In particular, the projection to F − ⁣ ∘ ∘ is in the image of the injection

H 1 ⁢ ( Q , F − ⁣ + + ⁢ D ⁢ ( f , g , h ) ∗ ) → H 1 ⁢ ( Q , F − ⁣ ∘ ∘ ⁢ D ⁢ ( f , g , h ) ∗ ) .

Then the reciprocity law of Bertolini–Seveso–Venerucci establishes a connection between the diagonal-cycle class and the 𝑝-adic 𝐿-function.

Theorem B3.4

The image of that class under the corresponding Coleman map recovers the 𝑝-adic 𝐿-function,

Col η f ⊗ ω g ⊗ ω h ⁡ ( loc p ⁡ ( κ ⁢ ( f , g , h ) ) ) = L p f ⁢ ( f , g , h ) .

Proof

This follows from [11, Theorem A]. Although the result in [11, Theorem A] is stated just for Hida families, the argument remains valid in the present generality, using the 𝑝-adic 𝐿-function of Andreatta–Iovita [1] and the general theory of ( φ , Γ ) -modules. ∎

B4 GL 1 over an imaginary quadratic field

B4.1 Setting

We fix an imaginary quadratic field 𝐾 where the prime 𝑝 splits. For simplicity, we shall always take 𝐾 to have class number one. We also fix embeddings

K ⊂ Q ̄ ↪ Q ̄ p

(determining a prime 𝔭 of 𝐾 above 𝑝).

We write 𝜎 and σ ̄ for the chosen embedding K ↪ Q p and its complex conjugate. If a − b = 0 mod w K : = # O K × , then the character σ a ⁢ σ ̄ b of K × is trivial on O K × , and thus determines an algebraic Grössencharacter of 𝐾 of conductor 1 (mapping a prime 𝔮 to λ a ⁢ λ ̄ b for 𝜆 any generator of 𝔮). We abuse notation slightly by writing σ a ⁢ σ ̄ b for this Grössencharacter. Its infinity type (with the conventions of [9] and [20, §2.1]) is ( a , b ) .

We let Σ K be the set of algebraic Grössencharacters of 𝐾. Define

Σ K crit = Σ K ( 1 ) ∪ Σ K ( 2 ) ⊂ Σ K

to be the disjoint union of the sets

Σ K ( 1 ) = { ξ ∈ Σ K ⁢ of infinity type ⁢ ( a , b ) , with ⁢ a ⩽ 0 , b ⩾ 1 } , Σ K ( 2 ) = { ξ ∈ Σ K ⁢ of infinity type ⁢ ( a , b ) , with ⁢ a ⩾ 1 , b ⩽ 0 } .

Thus ξ ∈ Σ K crit if and only if s = 0 is a critical point for L ⁢ ( ξ − 1 , s ) .

For Ψ any algebraic Grössencharacter of 𝐾, we may define a 1-dimensional Galois representation V p ⁢ ( Ψ ) , on which G K acts via the composite of Ψ with the Artin map (normalised to send geometric Frobenii to uniformisers, as in [20, §2.3.2]). In particular, we have V p ⁢ ( σ ⁢ σ ̄ ) ≅ Q p ⁢ ( − 1 ) , and L ⁢ ( V p ⁢ ( Ψ ) , s ) = L ⁢ ( Ψ , s ) .

B4.2 Character spaces

Let Γ K denote the group Gal ⁡ ( K ⁢ [ p ∞ ] / K ) , where K ⁢ [ p ∞ ] is the ray class field mod p ∞ , and let W K be the corresponding character space, so we have O ⁢ ( W K ) = H ⁢ ( Γ K ) . We let j K be the universal character Gal ⁡ ( K ab / K ) ↠ Γ K ↪ H ⁢ ( Γ K ) × .

We identify an algebraic Grössencharacter 𝜉 of 𝐾 unramified outside 𝑝 with the unique point of W K at which j K specialises to V p ⁢ ( ξ − 1 ) (note signs).

Remark B4.1

This inverse ensures that, if we identify Γ K with O K , p × / O K × via the restriction of the Artin map to O K , p × ⊂ A K , f × , the character x ↦ Nm K / Q ⁡ ( x ) corresponds to the cyclotomic character.

B4.3 Katz’s 𝑝-adic 𝐿-function

Let Ψ be a Grössencharacter of finite order and conductor coprime to 𝑝, with values in 𝐿. By the work of Katz [22] (see also [8, Theorem 3.1]), there exists a 𝑝-adic analytic function

L p Katz ⁢ ( Ψ ) : W K → L ⊗ Q p Q ̂ p nr ,

uniquely determined by the interpolation property that if ξ ∈ Σ K ( 2 ) is a character of conductor 1, hence necessarily of the form σ a ⁢ σ ̄ b for some a ⩾ 1 , b ⩽ 0 , then we have

L p Katz ⁢ ( Ψ ) ⁢ ( ξ ) Ω p a − b = a ⁢ ( ξ ) × e ⁢ ( ξ ) × f ⁢ ( ξ ) × L ⁢ ( Ψ ⁢ ξ − 1 , 0 ) Ω a − b ,

with both sides lying in Q ̄ , where

  1. a ⁢ ( ξ ) = ( a − 1 ) ! ⁢ π − b ,

  2. e ⁢ ( ξ ) = ( 1 − p − 1 ⁢ Ψ − 1 ⁢ ξ ⁢ ( p ) ) ⁢ ( 1 − ξ − 1 ⁢ Ψ ⁢ ( p ̄ ) ) ,

  3. f ⁢ ( ξ ) = D K b / 2 ⁢ 2 − b ,

  4. Ω p ∈ ( Q ̂ p nr ) × is a 𝑝-adic period attached to 𝐾,

  5. Ω ∈ C × is the complex period associated with 𝐾,

  6. L ⁢ ( Ψ ⁢ ξ − 1 , s ) is Hecke’s 𝐿-function associated with Ψ ⁢ ξ − 1 .

(More generally, one can state an interpolation property at all algebraic Grössencharacters ξ ∈ Σ K ( 2 ) of 𝑝-power conductor – not necessary of conductor 1 – but we shall not need this more general formula here.)

B4.4 Elliptic units

Again, let Ψ be a Grössencharacter of finite order and conductor coprime to 𝑝. Let ( − ) ∼ denote the reflexive hull of a H Γ -module. The Euler system of elliptic units can be thought as an element

κ ⁢ ( Ψ , K ) ∈ H 1 ⁢ ( K , V p ⁢ ( Ψ ) ∗ ⁢ ( 1 − j K ) ) ∼ ,

constructed by Coates and Wiles in their seminal paper [15]; see [21, §15] or [6, §1.2] for more recent accounts[3]. By construction, the specialisation of κ ⁢ ( Ψ , K ) at a finite-order character 𝜉 of Γ K is the image under the Kummer map of a linear combination of global units in an abelian extension of 𝐾.

Localising at 𝔭, and using the two-variable version of Perrin-Riou’s regulator defined in [30], we have a Coleman map

Col p , Ψ : H 1 ⁢ ( K p , V p ⁢ ( Ψ ) ∗ ⁢ ( 1 − j K ) ) → H ⁢ ( Γ K ) ,

which extends automatically to the reflexive hull. The explicit reciprocity law of Coates–Wiles links the system of elliptic units with Katz’s two variable 𝑝-adic 𝐿-function,

Col p , Ψ ⁡ ( loc p ⁡ κ ⁢ ( Ψ , K ) ) = L p Katz ⁢ ( Ψ ) .

B5 Heegner classes

We follow for this section the exposition of [20], which generalises Castella’s earlier results [14], and keep the notation of the previous sections. We suppose all primes dividing 𝑁 are split in 𝐾, and choose an ideal 𝔑 with O K / N ≅ Z / N . Let 𝐟 be a Coleman family of tame level 𝑁 defined over an affinoid disc V 1 .

In [20], we worked over an auxiliary rigid space V ̃ 1 (essentially a piece of the eigenvariety for GU ⁡ ( 1 , 1 ) ) parametrising conjugate-self-dual twists of the base-change of 𝐟 to 𝐾. To simplify the exposition, and since this is harmless towards our objectives of explaining the degeneration phenomena going on, we shall avoid appealing to this construction by make the following two simplifying assumptions:

  1. 𝐾 has class number one, as above, and moreover K ≠ Q ⁢ ( − 1 ) , Q ⁢ ( − 3 ) ;

  2. 𝐟 has trivial nebentype.

These simplifying hypotheses will allow us to split the parameter space up into two copies of 𝒲, one for the “weight” variable and one for the “anticyclotomic” one.

More precisely, assumption (b) allows us to choose a character m : Z p × → O ⁢ ( V 1 ) × which is a square root of the canonical character 𝐤, so 𝐟 has weight-character k + 2 = 2 ⁢ m + 2 . Slightly abusively, we write “ k 2 ” for this character. Hence the representation W = V ⁢ ( f ) ∗ ⁢ ( − k 2 ) of G Q satisfies W ≅ W ∗ ⁢ ( 1 ) .

Remark B5.1

Note that if 𝑘 is an integer in V 1 , then 𝑘 is necessarily even. However, the specialisation of “ k 2 ” at 𝑘 might not be x ↦ x k / 2 ; in fact, this holds only if 𝑘 lies in a certain congruence class mod 2 ⁢ ( p − 1 ) , lifting the congruence class mod ( p − 1 ) determined by the component of 𝒲 containing V 1 .

Meanwhile, assumption (a) implies that the Grössencharacter χ ac = σ / σ ̄ gives an isomorphism Γ ac ≅ Z p × , where Γ ac = O K , p × / Z p × is the Galois group of the maximal anticyclotomic extension unramified outside 𝑝. Composing this with the universal character

j : Z p × → O ⁢ ( W ) × ,

we obtain a universal anti-cyclotomic character j ac : G K → O ⁢ ( W ) × , whose specialisation at j ∈ W is the character mapping arithmetic Frobenius at a prime λ ∤ p of 𝐾 to ( σ ⁢ ( λ ) / σ ̄ ⁢ ( λ ) ) j .

Consider the module over O ⁢ ( V 1 × W ) defined by

V ac ( f ) ∗ : = V ( f ) ∗ ( − k 2 ) ⊗ ̂ L O ( W ) ( − j ac ) .

The Galois representation V ac ⁢ ( f ) ∗ is characterised by the property that, for any integers ( k , j ) , with k ∈ V 1 (and 𝑘 in the appropriate congruence class modulo 2 ⁢ ( p − 1 ) ), we recover the Galois representation

V ac ⁢ ( f ) ∗ ⁢ ( k , j ) = V p ⁢ ( f k × χ − 1 ) ∗ , χ = σ ( k / 2 + j ) ⁢ σ ̄ ( k / 2 − j ) ,

where f k is the weight k + 2 specialisation of 𝐟.

Remark B5.2

Note that ( f k , χ ) is a Heegner pair in the sense of [20]. This corresponds to the fact that the character χ ′ = χ ⋅ Nm is central critical for f k , in the sense of [9], so we have that L ⁢ ( f , χ − 1 , 1 ) = L ⁢ ( f , ( χ ′ ) − 1 , 0 ) is a central 𝐿-value; cf. [20, Remark 2.2.2] for the shift by 1. If − k 2 ⩽ j ⩽ k 2 , then χ ′ is a point of the region Σ cc ( 1 ) ⁢ ( f k ) in [9, Figure 1]. If j ⩾ k 2 + 1 , it is in Σ cc ( 2 ) ⁢ ( f k ) . (Similarly, it is in Σ cc ( 2 ′ ) ⁢ ( f ) if j ⩽ − 1 − k 2 , but we shall not consider this region here.)

In this scenario, we have

  • a Heegner class

    κ ⁢ ( f , K ) ∈ H 1 ⁢ ( K , V ac ⁢ ( f ) ∗ )

    constructed in [20, Theorem A], whose specialisations at ( k , j ) with − k 2 ⩽ j ⩽ k 2 are the Abel–Jacobi images of Heegner cycles;

  • an anticyclotomic 𝑝-adic 𝐿-function

    L p BDP ⁢ ( f ) ∈ O ⁢ ( V 1 × W )

    such that the square of its value at a point ( k , j ) with j ⩾ k 2 + 1 agrees with

    ( ∗ ) ⋅ L ⁢ ( f k / K × χ − 1 , 1 ) ,

    where ( ∗ ) is the usual combination of Euler factors;

  • an explicit reciprocity law [20, Theorem B]. To state this, we note that loc p ⁡ κ ⁢ ( f , K ) factors through the anticyclotomic Iwasawa cohomology of a rank 1 submodule

    F p + ⁢ D ⁢ ( f ) ∗ ⊂ D ⁢ ( f ) ∗ ,

    where D ⁢ ( f ) ∗ is the ( φ , Γ ) -module of V ⁢ ( f ) ∗ at 𝑝 (cf. [20, Theorem 6.3.4]). Let F p − ⁢ D ⁢ ( f ) denote the quotient of D ⁢ ( f ) dual to this; then there is a canonical basis vector ω f of D cris ⁢ ( F p − ⁢ D ⁢ ( f ) ) , interpolating the classes ω f for classical specialisations 𝑓 of 𝐟 (cf. Section A3.2), which we may use this to define a Coleman map Col p , ω f . The explicit reciprocity law then states that

    Col p , ω f ⁡ ( loc p ⁡ ( κ ⁢ ( f , K ) ) ) = ( − 1 ) ( k / 2 + j ) ⁢ L p BDP ⁢ ( f ) .

C Critical-slope Eisenstein specialisations

C1 Deformation of Beilinson–Flach elements

C1.1 Setup

In this section, we consider the Beilinson–Flach Euler system of [31] attached to two modular forms, or more generally to two Coleman families. This generalises the earlier construction of [24], where the variation was restricted to the case of ordinary families.

We review some of the notation already introduced in previous parts to make the section more self-contained. Let f = E r + 2 ⁢ ( ψ , τ ) stand for the Eisenstein series of weight r + 2 and characters ( ψ , τ ) , and f β its critical-slope 𝑝-stabilisation. Under the non-criticality conditions discussed in Theorem A4.5, there is a unique Coleman family 𝐟 passing through f β , over some affinoid disc V 1 ∋ r . We may suppose that, for all integers k ∈ V 1 ∩ Z ⩾ 0 with k ≠ r , the specialisation f k is a non-critical-slope cusp form.

Meanwhile, let 𝐠 be a second Coleman family over some disc V 2 . We suppose for simplicity that 𝐠 is ordinary. Let V ⁢ ( f , g ) ∗ be the module defined in Section B3.1. Recall that the Galois representation V ⁢ ( f , g ) ∗ is characterised by the property that, for any integers ( k , ℓ , j ) , with k ∈ V 1 and ℓ ∈ V 2 , we recover

V ⁢ ( f , g ) ∗ ⁢ ( k , ℓ , j ) = V ⁢ ( f k ) ∗ ⊗ V ⁢ ( g ℓ ) ∗ ⁢ ( − j ) ,

the ( − j ) -th Tate twist of the tensor product of the dual Galois representations attached to f m and g ℓ , as defined in Definition A6.2 (including the case k = r , when f k = f β ). We define similarly a space V c ⁢ ( f , g ) ∗ using V c ⁢ ( f ) ∗ in place of V ⁢ ( f ) ∗ . Note that these become isomorphic after inverting 𝑋.

C1.2 Selmer vanishing

The representation V ⁢ ( g ) ∗ has a canonical rank-one G Q p -subrepresentation F + ⁢ V ⁢ ( g ) ∗ , with unramified quotient F − ⁢ V ⁢ ( g ) ∗ , and we have the following.

Proposition C1.1

For any integer 𝑛 and Dirichlet character 𝜒, the “Greenberg Selmer group”

H Gr 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⁢ ( χ ) ⁢ ( n ) ⊗ H Γ ⁢ ( − j ) ) := ker ⁡ ( H 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⁢ ( χ ) ⁢ ( n ) ⊗ H Γ ⁢ ( − j ) ) → H 1 ⁢ ( Q p , F − ⁢ V ⁢ ( g ) ∗ ⁢ ( χ ) ⁢ ( n ) ⊗ H Γ ⁢ ( − j ) ) )

vanishes.

Proof

We may take n = 0 and χ = 1 without loss of generality. The result now follows from Kato’s theorems [21], which show that, for each classical specialisation g ℓ of 𝐠, the module H 1 ⁢ ( Q , V ⁢ ( g ℓ ) ∗ ⊗ H Γ ⁢ ( − j ) ) is free of rank 1 over H Γ and contains a canonical element (Kato’s Euler system for g ℓ ) whose localisation at 𝑝 is mapped to the 𝑝-adic 𝐿-function of g ℓ under the Perrin-Riou regulator for F − ⁢ V ⁢ ( g ℓ ) ∗ . Since the 𝑝-adic 𝐿-function is not a zero-divisor, we conclude that the space H Gr 1 ⁢ ( Q , V ⁢ ( g ℓ ) ∗ ⊗ H Γ ⁢ ( − j ) ) vanishes for each such g ℓ .

So any element of H Gr 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⊗ H Γ ⁢ ( − j ) ) must specialise to 0 at a Zariski-dense set of points of V 2 . On the other hand, this module is contained in the full H 1 , which is O ⁢ ( V 2 ) -torsion-free, by the exact sequence associated to multiplication by an element of O ⁢ ( V 2 ) . Hence the Greenberg Selmer group vanishes. ∎

Remark C1.2

It is slightly irritating that our analysis of the specialisation of Beilinson–Flach elements relies on these Selmer-group bounds, and thus on the existence of Kato’s Euler system. This would be an obstacle if we wanted to use the techniques of “critical-slope Eisenstein specialisations” to define new Euler systems (rather than obtaining relations between existing Euler systems).

C1.3 Families over punctured discs

Proposition C1.3

The cohomology H 1 ⁢ ( Q , V ⁢ ( f , g ) ∗ ) is finitely generated over

O ⁢ ( V 1 × V 2 × W ) ,

and this module is 𝑋-torsion-free, where X ∈ O ⁢ ( V 1 ) is a uniformiser at 𝑟.

Proof

This follows, via the exact sequence of cohomology associated to multiplication by 𝑋, from the vanishing of H 0 ⁢ ( Q , V ⁢ ( f , g ) ∗ / X ) = H 0 ⁢ ( Q , V ⁢ ( f β ) ∗ ⊗ V ⁢ ( g ) ∗ ⊗ H Γ ⁢ ( − j ) ) . ∎

Theorem C1.4

Fix d ∈ Z > 1 such that ( d , 6 ⁢ S ) = 1 . There exists a cohomology class

κ d ⁢ ( f , g ) ∈ H 1 ⁢ ( Q , 1 X ⁢ V c ⁢ ( f , g ) ∗ ) ,

with the following interpolation property: if ( k , ℓ ) are integers at least 0 with k ≠ r , then we have

κ d ⁢ ( f , g ) ⁢ ( k , ℓ ) = B d ⁢ F [ f k , g ℓ ] ∈ H 1 ⁢ ( Q , V ⁢ ( f k ) ∗ ⊗ V ⁢ ( g ℓ ) ∗ ⊗ H Γ ⁢ ( − j ) ) ,

where the element B d ⁢ F [ f k , g ℓ ] = B d ⁢ F 1 , 1 [ f k , g ℓ ] is as defined in [31, Theorem 3.5.9].

Proof

Compare [31, Theorem 5.4.2], which is an analogous result when all integer-weight specialisations of 𝐠 are classical. In the present situation, we must be a little more circumspect, since [31, Proposition 5.2.5] does not apply for k = r , so the map denoted pr g [ j ] there is not defined for j = k + 1 . However, inverting 𝑋 gets rid of this problem. ∎

C1.4 Local properties at 𝑝

Proposition C1.5

The image of loc p ⁡ ( κ d ⁢ ( f , g ) ) in H 1 ⁢ ( Q p , 1 X ⁢ F − − ⁢ D c ⁢ ( f , g ) ∗ ) is zero.

Proof

This follows from the fact that the Iwasawa cohomology is torsion-free, and the specialisations away from X = 0 have the required vanishing property. ∎

C1.5 Leading terms at X = 0

We now consider the quotient

1 X ⁢ V c ⁢ ( f , g ) ∗ V ⁢ ( f , g ) ∗ ,

which makes sense by the discussion of Section A6.3 leading to the chain of inclusions (A6.1).

Proposition C1.6

The image of κ d ⁢ ( f , g ) in the cohomology of the quotient

1 X ⁢ V c ⁢ ( f , g ) ∗ V ⁢ ( f , g ) ∗ ≅ Q p ⁢ ( τ − 1 ) ⁢ ( 1 + r ) ⊗ V ⁢ ( g ) ∗ ⊗ H Γ ⁢ ( − j )

is zero.

Proof

Firstly, observe that the isomorphism of the statement follows from Section A6.3. We consider the projection of this class to the local cohomology at 𝑝 of the quotient F − ⁢ V ⁢ ( g ) ∗ . Since the ( φ , Γ ) -module of V c ⁢ ( f β ) quo ∗ injects into F − ⁢ D c ⁢ ( f β ) , this projection is 0, by Proposition C1.1. Hence the global class lands in the Greenberg Selmer group, which is zero, as we have seen. ∎

Corollary C1.7

The class κ d ⁢ ( f , g ) lifts (uniquely) to H 1 ⁢ ( Q , V ⁢ ( f , g ) ∗ ) , and thus has a well-defined image

κ ̂ d ⁢ ( f β , g ) ∈ H 1 ⁢ ( Q , Q p ⁢ ( ψ − 1 ) ⊗ V ⁢ ( g ) ∗ ⊗ H Γ ⁢ ( − j ) ) .

For the following result, recall the logarithmic distribution, as introduced for instance in [5, §1]: for a continuous character σ : Z p × → C p , the function

d k ⁢ σ d ⁢ z k ⋅ z k σ ⁢ ( z )

is a constant, and log [ k ] ∈ C p is defined to be this constant.

Proposition C1.8

The class κ ̂ d ⁢ ( f β , g ) is divisible by the (cyclotomic) logarithm distribution log [ r + 1 ] ∈ H Γ .

Proof

The specialisation of κ d ⁢ ( f , g ) at any locally algebraic character of Γ whose degree 𝑗 lies in { 0 , … , r } factors through the image of D U − j ⊗ TSym j in

D U − ( r + 1 ) ⊗ TSym ( r + 1 ) ,

and the maps Pr f [ j ] and Pr f [ r + 1 ] agree on this image up to a non-zero scalar. Since the Pr f [ j ] for 0 ⩽ j ⩽ r do not have poles at X = 0 , it follows that the specialisations of κ d ⁢ ( f , g ) at triples ( r , ℓ , χ ) , for ℓ ⩾ r and 𝜒 locally algebraic of degree ∈ { 0 , … , r } , interpolate the projections of the classical Beilinson–Flach classes to the ( E r + 2 crit , g ℓ ) -eigenspaces in classical cohomology. Since the Beilinson–Flach classes arise as suitable projections of classes in the cohomology of X 1 ⁢ ( N ) × Y 1 ⁢ ( N ) , these projections are always 0. By Zariski-density, the class specialises to 0 everywhere in { r } × V 2 × { χ } . Since this holds for all 𝜒 of degree up to 𝑟, and these are exactly the zeroes of log [ r + 1 ] , the result follows. ∎

Since the Iwasawa cohomology is torsion-free, there is a unique class

κ d ⁢ ( f β , g ) ∈ H 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⁢ ( ψ − 1 ) ⊗ H Γ ⁢ ( − j ) )

such that

κ ̂ d ⁢ ( f β , g ) = log [ r + 1 ] ⋅ κ d ⁢ ( f β , g ) .

Moreover, since 𝐠 is ordinary, the class κ d ⁢ ( f β , g ) has bounded growth and hence lies in the submodule H 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⁢ ( ψ − 1 ) ⊗ Λ Γ ⁢ ( − j ) ) , where Λ Γ is the cyclotomic Iwasawa algebra. (More generally, we could carry this out with a non-ordinary family 𝐠, and we would obtain a class with growth of order equal to the slope of 𝐠.)

C1.6 The 𝑝-adic 𝐿-function

Consider the “𝐠-dominant” 𝑝-adic 𝐿-function L p g ⁢ ( f , g ) over V 1 × V 2 × W . The interpolation property applies also at k = r without any special complications, and here the complex 𝐿-function factors as

(C1.1) L ⁢ ( E r + 2 ⁢ ( ψ , τ ) , g ℓ , 1 + j ) = L ⁢ ( g ℓ , ψ , 1 + j ) ⋅ L ⁢ ( g ℓ , τ , j − r ) .

Note that both factors on the right-hand side are critical values. Thus the restriction of L p g ⁢ ( f , g ) to the k = r fibre is uniquely determined by its interpolation property at crystalline points (we do not need finite-order twists), and we have an “Artin formalism” factorisation

L p g ⁢ ( E r + 2 ⁢ ( ψ , τ ) , g ) ⁢ ( j ) = L p ⁢ ( g × ψ , j ) ⋅ L p ⁢ ( g × τ , j − 1 − r ) L p ⁢ ( Ad ⁡ g ) ,

where the denominator arises from the choice of periods.

C1.7 Perrin-Riou maps

We want to relate L p g ⁢ ( f , g ) to the image of loc p ⁡ ( κ d ⁢ ( f , g ) ) under the projection to F − ⁢ V ⁢ ( g ) ∗ . As discussed in Section B3.1, this factors through the natural map

H Iw 1 ⁢ ( Q p , ∞ , F + − ⁢ D ⁢ ( f , g ) ∗ ) → H Iw 1 ⁢ ( Q p , ∞ , F ∘ − ⁢ D ⁢ ( f , g ) ∗ ) ,

which is injective (since H Iw 0 ⁢ ( F − − ) will be zero). Perrin-Riou’s regulator gives us a map

Col b f + ⊗ η g = ⟨ L F + − PR ⁢ ( − ) , b f + ⊗ η g ⟩ : H Iw 1 ⁢ ( Q p , ∞ , F + − ⁢ D ⁢ ( f , g ) ∗ ) → O ⁢ ( V 1 × V 2 × W )

which interpolates the Perrin-Riou regulators for f k × g ℓ for varying ( k , ℓ ) .

Let φ − 1 stand for the left inverse of the Frobenius, denoted as 𝜓 in [24, §8.2]. More precisely, proceeding as in [24, §8.2] and using Fontaine isomorphism, for

z ∈ ( F + − ⁢ D ⁢ ( f , g ) ∗ ) φ − 1 = 1 ,

this map sends 𝑧 to

⟨ ι ⁢ ( ( 1 − φ ) ⁢ z ) , b f + ⊗ η g ⟩ ,

where 𝜄 is the inclusion of ( F + − ⁢ D ⁢ ( f , g ) ∗ ) φ − 1 = 0 into the product

( F + − ⁢ D ⁢ ( f , g ) ∗ ⁢ [ 1 / t ] ) φ − 1 = 0 = D cris ⁢ ( F + ⁢ D ⁢ ( f ) ∗ ⁢ ( − 1 − k ) ) ⊗ D cris ⁢ ( F − ⁢ D ⁢ ( g ) ∗ ) ⊗ H Γ .

For the following discussion, recall the constants introduced in Section A8.3. At the bad fibre, we have the relation

b f + mod X = c r ⁢ t r + 1 ⁢ η f r α ⊗ e − ( r + 1 ) ,

where e n is the standard basis of Z p ⁢ ( n ) and c r is a nonzero constant. Since multiplication by t r + 1 corresponds to multiplication by log [ r + 1 ] on the H Γ side, we conclude that

Col b f + ⊗ η g ⁡ ( κ d ⁢ ( f , g ) ) mod X = c r ⁢ ⟨ L F − ⁢ V ⁢ ( g ) ∗ ⁢ ( ψ − 1 ) PR ⁢ ( κ d ⁢ ( E ℓ + 2 crit , g ) ) , η f k α ⊗ η g ⟩ .

Theorem C1.9

We have

Col b f + ⊗ η g ⁡ ( κ d ⁢ ( f , g ) ) = c f ⁢ ( k ) ⋅ C d ⁢ ( f , g , j ) ⋅ L p g ⁢ ( f , g ) ,

where c f ⁢ ( k ) is a meromorphic function on V 1 alone, regular and non-vanishing at all integer weights k ⩾ − 1 , except possibly at k = r , where it is regular.

Proof

It follows easily from the reciprocity laws of Theorem B3.4 that the quotient

Col b f + ⊗ η g ⁡ ( κ d ⁢ ( f , g ) ) / ( C d ⁢ ( f , g , j ) ⁢ L p g ⁢ ( f , g ) )

is a function of 𝐤 alone, and this ratio does not vanish at any integer k ⩾ − 1 where g k is classical and cuspidal; it is equal to the fudge factor c k defined above using the results of Section A8.3.

Moreover, since L p g ⁢ ( f , g ) is well-defined and non-zero along { r } × V 2 × W , we conclude that c f ⁢ ( k ) does not have a pole at 𝐤 (although it might have a zero there). ∎

C1.8 Meromorphic Eichler–Shimura

Theorem C1.10

There exist an integer n ⩾ 0 and a unique isomorphism of O ⁢ ( V 1 ) -modules

ω f : D cris ⁢ ( F + ⁢ D ⁢ ( f ) ∗ ⁢ ( − 1 − k ) ) ≅ X − n ⁢ O ⁢ ( V 1 ) ,

whose specialisation at every k ⩾ 1 ∈ V 1 with k ≠ r is the linear functional given by pairing with the differential form ω f k associated to the weight k + 2 specialisation of 𝐟. For this ω f , we have

⟨ L F + − PR ⁢ ( κ d ⁢ ( f , g ) ) , ω f + ⊗ η g ⟩ = C d ⁢ ( f , g , j ) ⋅ L p g ⁢ ( f , g ) .

Proof

We simply define ω f to be the quotient b f + / c f , and 𝑛 the order of vanishing of c f at k = r . ∎

Remark C1.11

Note that we used the family 𝐠 in the construction of ω f , but the interpolating property relating it to the classical Eichler–Shimura isomorphism implies that it is uniquely determined by 𝐟 alone.

C1.9 Leading terms when c f ⁢ ( r ) = 0

If c f ⁢ ( r ) ≠ 0 , then we have thus constructed a class in Iwasawa cohomology of V ⁢ ( g × ψ ) ∗ whose regulator agrees with the product of Kato’s Euler system for g × ψ , and a shifted copy of the 𝑝-adic 𝐿-function for g × τ .

We claim that if c f ⁢ ( r ) = 0 , then in fact κ d ⁢ ( g , h ) is divisible by 𝑋. If c f ⁢ ( r ) = 0 , then κ d ⁢ ( E ℓ + 2 crit , g ) is in the Selmer group with local condition F + ⁢ V ⁢ ( g ) ∗ . This Selmer group is 0 by Proposition C1.1. So κ d ⁢ ( g , h ) mod X would have to land in the cohomology of V ⁢ ( g ) sub ∗ instead, but then we are seeing the projection into F − , not F + , so by Kato’s results again (for g × τ , instead of g × ψ , this time), this is zero as well.

So we can divide out a factor of 𝑋 from both κ d ⁢ ( f , g ) and c f ⁢ ( k ) , and repeat the argument. Since c f is not identically 0, this must terminate after finitely many steps. Thus we have shown the following.

Proposition C1.12

If n ⩾ 0 is the order of vanishing of c f at k = r , then X − n ⁢ κ ̂ d ⁢ ( f , g ) is well-defined and non-zero modulo 𝑋, and this leading term projects non-trivially into the quotient H 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⁢ ( ψ − 1 ) ⊗ H Γ ⁢ ( − j ) ) . Its image under the Perrin-Riou regulator is given by

c f ∗ ⁢ ( r ) ⁢ C d ⁢ ( f β , g , j ) ⋅ log [ r + 1 ] ⋅ L p g ⁢ ( E r + 2 ⁢ ( ψ , τ ) , g ) ,

where c f ∗ ⁢ ( r ) ∈ L × .

We denote the resulting class by κ ̂ ∗ d ⁢ ( f β , g ) , and define κ ∗ d ⁢ ( f β , g ) in the same way using κ d ⁢ ( f β , g ) instead. If n = 0 , we have seen above that this class is divisible by log [ r + 1 ] ; for n > 0 , this is less obvious, but it follows from the proof of the next theorem.

Theorem C1.13

We have

κ ̂ ∗ d ⁢ ( f β , g ) = ( C ⋅ C d ⁢ ( f β , g , j ) ⁢ log [ r + 1 ] ⋅ L p ⁢ ( g ⊗ τ , j − 1 − r ) ) L p ⁢ ( Ad ⁡ g ) ⋅ κ ⁢ ( g × ψ )

for some nonzero constant C ∈ L × .

Proof

Taking C = c f ∗ ⁢ ( r ) , it follows from equation (C1.1) and the previous proposition (combined with Kato’s reciprocity law for 𝐠) that both of the cohomology classes we are considering have the same image under the regulator, so they are equal as cohomology classes, by Proposition C1.1. ∎

Remark C1.14

In terms of the class κ ∗ d ⁢ ( f β , g ) , which already includes the logarithmic factor, the result takes the easier form of

κ ∗ d ⁢ ( f β , g ) = ( C ⋅ C d ⁢ ( f β , g , j ) ⋅ L p ⁢ ( g ⊗ τ , j − 1 − r ) ) L p ⁢ ( Ad ⁡ g ) ⋅ κ ⁢ ( g × ψ ) .

C2 Deformation of diagonal cycles

C2.1 Setup

In this section, we consider the diagonal cycles of [11] attached to three modular forms, or more generally to three Coleman families.

Let f = E r + 2 ⁢ ( ψ , τ ) stand for the Eisenstein series of weight r + 2 and characters ( ψ , τ ) , with ψ ⁢ τ = χ f . As before, we take its critical-slope Eisenstein 𝑝-stabilisation, that we denote as E r + 2 crit ⁢ ( ψ , τ ) or just E k + 2 crit if the choice of characters is clear from the context. Let ( g , h ) be two modular forms of weights ( ℓ + 2 , m + 2 ) , levels ( N g , N h ) , and nebentypes ( χ g , χ h ) . We make the self-duality assumption χ f ⁢ χ g ⁢ χ h = 1 , and to simplify notation, suppose that ℓ ⩾ m . We further fix 𝑝-stabilisations of 𝑔 and ℎ, that we denote as g α and h α respectively. Under the non-criticality conditions already discussed, we may fix a triple of Coleman families ( f , g , h ) passing through ( E r + 2 crit , g α , h α ) over a triple of affinoid discs ( V 1 , V 2 , V 3 ) . For simplicity, we may assume that both 𝐠 and 𝐡 are ordinary families and, as before, that, for all integers k ∈ V 1 ∩ Z ⩾ 0 with k ≠ r , the specialisation f k is a non-critical slope cusp form.

As in Section B3.2 above, we choose a value of k 1 + k 2 + k 3 2 , as a family of characters over the product V 1 × V 2 × V 3 , and we say a triple of integer weights ( k , ℓ , m ) is an “integer point” if it is compatible with this choice of square roots.

We want to consider the diagonal class attached by the works of Darmon–Rotger [17] and Bertolini–Seveso–Venerucci [11] to the triple ( f , g , h ) , that we denote by κ ⁢ ( f , g , h ) . Recall the module V ⁢ ( f , g , h ) ∗ defined in (B3.1). We define similarly a space V c ⁢ ( f , g , h ) ∗ using V c ⁢ ( f ) ∗ instead of V ⁢ ( f ) ∗ .

C2.2 Selmer vanishing

With the previous notation, consider the family of representations over O ⁢ ( V 2 × V 3 ) given by

V ( g , h ) 0 ∗ : = ( V ( g ) ∗ ⊗ ̂ Q p V ( h ) ∗ ) ( − 1 − r + k 2 + k 3 2 ) .

Here r + k 2 + k 3 2 is understood as a character of Z p × via our choice above specialised at k 1 = r . This has a rank 1 submodule

F + + ⁢ V ⁢ ( g , h ) 0 ∗ = ( F + ⁢ V ⁢ ( g ) ∗ ⊗ ̂ Q p F + ⁢ V ⁢ ( h ) ∗ ) ⁢ ( − 1 − r + k 2 + k 3 2 ) .

Let 𝑛 be an integer number, playing the role of a Tate twist (later, we will take 𝑛 to be either 0 or 1 + r ). We consider the two-variable 𝑝-adic 𝐿-functions L p g ⁢ ( g , h ) and L p h ⁢ ( g , h ) restricted to s = 2 + r + k 2 + k 3 2 − n , which are analytic functions on V 2 × V 3 . We denote these functions as L p g ⁢ ( g , h ) | n and L p h ⁢ ( g , h ) | n respectively.

Lemma C2.1

Let n ≠ r + 1 2 . Then the 𝑝-adic 𝐿-functions L p g ⁢ ( g , h ) | n and L p h ⁢ ( g , h ) | n are non-zero.

Proof

These are two-variable 𝑝-adic 𝐿-functions depending on the two-weight variable, and interpolating the cyclotomic twist corresponding to a translation of t = r + 1 2 − n of the central value, which is ℓ + m + 3 2 . If t ⩾ 1 , the non-vanishing follows from the convergence of the Euler product. The case t = 1 2 follows from results of Shahidi [36, Theorem 5.2] on non-vanishing of 𝐿-functions for GL n on the abscissa of convergence. ∎

Proposition C2.2

For any integer n ≠ r + 1 2 and prime-to-𝑝 Dirichlet character 𝜒, the “Greenberg Selmer group”

H + + 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) := ker ⁡ ( H 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) → H 1 ⁢ ( Q p , V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) H 1 ⁢ ( Q p , F + + ⁢ ( V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) ) )

vanishes.

Proof

We can arrange χ = 1 without loss of generality. We shall compare the Greenberg Selmer group above (defined by a “codimension 3” local condition) with the Selmer group defined by a less restrictive “codimension 2” local condition,

H ∘ + 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) := ker ⁡ ( H 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) → H 1 ⁢ ( Q p , V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) H 1 ⁢ ( Q p , F ∘ + ⁢ ( V ⁢ ( g , h ) 0 ∗ ⁢ ( χ ) ⁢ ( n ) ) ) ) ,

where

F ∘ + ⁢ V ⁢ ( g , h ) 0 ∗ = ( V ⁢ ( g ) ∗ ⊗ ̂ Q p F + ⁢ V ⁢ ( h ) ∗ ) ⁢ ( − 1 − r + k 2 + k 3 2 ) .

If ( x , y ) is any point (not necessarily classical) of V 1 × V 2 at which L p g ⁢ ( g , h ) | n does not vanish, then the theory of Beilinson–Flach elements shows that H ∘ + 1 ⁢ ( Q , V ⁢ ( g x , h y ) ∗ ⁢ ( n ) ) is zero, and hence a fortiori so is H + + 1 ⁢ ( Q , V ⁢ ( g x , h y ) ∗ ⁢ ( n ) ) . Hence any element of H + + 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( n ) ) must specialise to 0 at a Zariski-dense set of points of V 2 × V 3 . On the other hand, this module is contained in the full H 1 , which is O ⁢ ( V 2 × V 3 ) -torsion-free, by an argument similar to the previous section. So H + + 1 is the zero module. ∎

C2.3 Families over punctured discs

As before, we have a freeness result.

Proposition C2.3

The cohomology H 1 ⁢ ( Q , V ⁢ ( f , g , h ) ∗ ) is finitely generated as a module over O ⁢ ( V 1 × V 2 × V 3 ) , and this module is 𝑋-torsion-free, where X ∈ O ⁢ ( V 1 ) is a uniformiser at 𝑟.

Proof

Similar to Proposition C1.3. ∎

Proposition C2.4

There exists a cohomology class

κ ⁢ ( f , g , h ) ∈ H 1 ⁢ ( Q , 1 X ⁢ V c ⁢ ( f , g , h ) ∗ ) ,

whose fibre at any balanced integer point ( k , ℓ , m ) with k ≠ r is the diagonal-cycle class of Section B3.2.

Proof

As noted in Remark B3.3, the construction of cohomology classes [11, §8] does not quite work in the present setting, because the Ohta pairing V ⁢ ( f ) × V c ⁢ ( f ) ∗ → O ⁢ ( V 1 ) used in [11, equation (82)] is not perfect. However, we have shown above that the Ohta pairing does induce a perfect duality between V c ⁢ ( f ) and 1 X ⁢ V c ⁢ ( f ) ∗ (for small enough V 1 ), and substituting this statement for the erroneous claim in [11], we obtain a cohomology class valued in 1 X ⁢ V c ⁢ ( f , g , h ) ∗ interpolating the diagonal-cycle classes for classical specialisations. ∎

Remark C2.5

For any affinoid subdomain V 1 ′ ⊂ V 1 not containing 𝑟, the construction of [11] does apply over V 1 ′ , and the restriction of our class κ ⁢ ( f , g , h ) to V 1 ′ × V 2 × V 3 is the diagonal-cycle cohomology class κ ⁢ ( f | V 1 ′ , g , h ) of [11].

C2.4 Local properties at 𝑝

Consider the rank 4 submodule

F bal + ⁢ D c ⁢ ( f , g , h ) ∗ = ( F + D c ( f ) ∗ ⊗ ̂ F + D ( g ) ∗ ⊗ ̂ D ( h ) ∗ + F + ⁢ D c ⁢ ( f ) ∗ ⊗ ̂ D ⁢ ( g ) ∗ ⊗ ̂ F + ⁢ D ⁢ ( h ) ∗ + D c ( f ) ∗ ⊗ ̂ F + D ( g ) ∗ ⊗ ̂ F + D ( h ) ∗ ) ( − 1 − k 1 + k 2 + k 3 2 ) .

We also consider the quotient

F bal − ⁢ D c ⁢ ( f , g , h ) ∗ = D c ⁢ ( f , g , h ) ∗ F + ⁢ D c ⁢ ( f , g , h ) ∗ .

By construction, for weights in the balanced region, the submodule F bal + satisfies the Panchishkin condition (i.e. all its Hodge–Tate weights are at least 1, and those of the quotient are at most 0).

Proposition C2.6

The image of loc p ⁡ ( κ ⁢ ( f , g , h ) ) in H 1 ⁢ ( Q p , 1 X ⁢ F bal − ⁢ D c ⁢ ( f , g , h ) ∗ ) is zero.

Proof

This follows from the fact that the Galois module is torsion-free, and the specialisations away from X = 0 have the required vanishing property (as they are built from cohomology classes which satisfy the Bloch–Kato local condition). ∎

C2.5 Specialisation at X = 0

Proposition C2.7

The image of κ ⁢ ( f , g , h ) in the cohomology of the quotient

1 X ⁢ V c ⁢ ( f , g , h ) ∗ V ⁢ ( f , g , h ) ∗ ≅ V ⁢ ( g , h ) 0 ∗ ⁢ ( τ − 1 ) ⁢ ( 1 + r )

is zero.

Proof

The image of the balanced submodule F bal + in this quotient is exactly the local condition defining the Greenberg Selmer group H + + 1 considered above (with χ = τ − 1 and n = 1 + r ). By Proposition C2.2, the Selmer group with this local condition is zero. Hence κ ⁢ ( f , g , h ) must map to the zero class in this module. ∎

Corollary C2.8

The class κ ⁢ ( f , g , h ) lifts (uniquely) to H 1 ⁢ ( Q , V ⁢ ( f , g , h ) ∗ ) , and thus has a well defined image in the module

κ ̂ ⁢ ( f β , g , h ) ∈ H 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( ψ − 1 ) ) .

Proposition C2.9

The cohomology class κ ̂ ⁢ ( f β , g , h ) is divisible by the logarithmic distribution log [ r + 1 ] ⁡ ( r − k 2 + k 3 2 ) .

Proof

We identify weights with quadruples ( k , ℓ , m , j ) with k + ℓ + m = 2 ⁢ j . We claim that the class κ ⁢ ( f , g , h ) vanishes at ( r , ℓ , m , j ) for all ℓ , m ⩾ 0 such that ( r , ℓ , m ) is balanced, i.e. | ℓ − m | ⩽ r with ℓ + m + r even. Indeed, the specialisation of κ ⁢ ( f , g , h ) at one such point factors through the image of D U − j ⊗ TSym j in D U − ( r + 1 ) ⊗ TSym ( r + 1 ) , and the maps Pr f [ j ] and Pr f [ r + 1 ] agree on this image up to a non-zero scalar.

Since Pr f [ j ] for 0 ⩽ j ⩽ r do not have poles at X = 0 , it follows that the specialisations of κ ⁢ ( f , g , h ) at triples ( r , ℓ , m , χ ) , for | ℓ − m | ⩽ r , ℓ + m + r even and 𝜒 locally algebraic of degree ∈ { 0 , … , r } , interpolate the projections of the diagonal cycles to the ( E r + 2 crit , g ℓ , h m ) -eigenspaces in classical cohomology. Since the diagonal classes lift to

X 1 ⁢ ( N ) × Y 1 ⁢ ( N ) × Y 1 ⁢ ( N ) ,

these projections are always 0. By Zariski-density, the class specialises to 0 everywhere in

( { r } × V 2 × V 3 ) ∩ ( | ℓ − m | ⩽ r )

with ℓ + m + r even, and the desired divisibility follows. ∎

Since the Iwasawa cohomology is torsion-free, there is a unique class

κ ⁢ ( f β , g , h ) ∈ H 1 ⁢ ( Q , V ⁢ ( g , h ) 0 ∗ ⁢ ( ψ − 1 ) )

such that

κ ̂ ⁢ ( f β , g , h ) = log [ r + 1 ] ⁡ ( r − k 2 + k 3 2 ) ⋅ κ ⁢ ( f β , g , h ) .

Proposition C2.10

This class κ ⁢ ( f β , g , h ) maps to 0 in the cohomology of the rank-one quotient Q p ⁢ ( ψ − 1 ) ⊗ F − ⁢ V ⁢ ( g ) ∗ ⊗ F − ⁢ V ⁢ ( h ) ∗ .

Proof

Since κ ⁢ ( f , g , h ) lies in the balanced Selmer group by construction, its image in the quotient V ⁢ ( f ) ∗ ⊗ F − ⁢ V ⁢ ( g ) ∗ ⊗ F − ⁢ V ⁢ ( h ) ∗ vanishes identically over V 1 × V 2 × V 3 . So it is in particular zero when we specialise at k = r . ∎

C2.6 The 𝑝-adic 𝐿-function

We assume for the rest of this section that the tame level of the Eisenstein series is trivial. For the construction of the triple product 𝑝-adic 𝐿-function, the interpolation property also applies at k = r . Because of the functional equation for the Rankin 𝐿-function and our assumption on the tame level, there is an equality

L ⁢ ( E r + 2 ⁢ ( ψ , τ ) , g ℓ , h m , 2 + r + ℓ + m 2 ) = L ⁢ ( g ℓ , h m × ψ , 2 + r + ℓ + m 2 ) 2 ,

where we have used that ψ ⁢ τ ⁢ χ g ⁢ χ h = 1 . Observe that the restriction of L p g ⁢ ( f , g , h ) to the region defined by k = r is uniquely determined by the interpolation property at crystalline points, and we have then an equality of 𝑝-adic 𝐿-functions

L p g ⁢ ( E r + 2 ⁢ ( ψ , τ ) , g , h ) = L p g ⁢ ( g , h × ψ , 2 + r + k 2 + k 3 2 ) .

C2.7 Perrin-Riou maps

We can relate the previous 𝑝-adic 𝐿-function to the image of the class loc p ⁡ ( κ ⁢ ( f , g , h ) ) under the projection to F − ⁢ V ⁢ ( g ) ∗ ⊗ F + ⁢ V ⁢ ( h ) ∗ . More precisely, Perrin-Riou’s regulator gives us a map

Col b f + ⊗ η g ⊗ ω h = ⟨ L F + ⁣ − + PR ⁢ ( − ) , b f + ⊗ η g ⊗ ω h ⟩ : H 1 ⁢ ( Q p , F + ⁣ − + ⁢ D ⁢ ( f , g , h ) ∗ ) → O ⁢ ( V 1 × V 2 × V 3 )

which interpolates the Perrin-Riou regulators for f k ⊗ g ℓ ⊗ h m . For

z ∈ ( F + ⁣ − + ⁢ D ⁢ ( f , g , h ) ∗ ) φ − 1 = 1 ,

this map sends 𝑧 to

⟨ ι ⁢ ( ( 1 − φ ) ⁢ z ) , b f + ⊗ η g ⊗ ω h ⟩ ,

where 𝜄 is now the inclusion of ( F + ⁣ − + ⁢ D ⁢ ( f , g , h ) ∗ ) φ − 1 = 0 into

D cris ⁢ ( F + ⁢ D ⁢ ( f ) ∗ ⁢ ( − 1 − k 1 ) ) ⊗ D cris ⁢ ( F − ⁢ D ⁢ ( g ) ∗ ) ⊗ D cris ⁢ ( F + ⁢ D ⁢ ( h ) ∗ ⁢ ( − 1 − k 3 ) ) .

Proceeding as with Beilinson–Flach classes, we conclude that

Col b f + ⊗ η g ⊗ ω h ⁡ ( κ ⁢ ( f , g , h ) ) mod X = c r ⁢ ⟨ L F − + ⁢ V ⁢ ( g ) ∗ ⊗ ̂ Q p V ⁢ ( h ) ∗ ⁢ ( ψ − 1 ) PR ⁢ ( κ ⁢ ( E r + 2 crit , g , h ) ) , η f r α ⊗ η g ⊗ ω h ⟩ .

The following result follows from the reciprocity law of [11], with the obvious modifications to adapt it to the Coleman case, exactly as in [31].

Theorem C2.11

We have

Col b f + ⊗ η g ⊗ ω h ⁡ ( κ ⁢ ( f , g , h ) ) = c ⁢ ( k ) ⋅ L p g ⁢ ( f , g , h ) ,

where c ⁢ ( k ) is a meromorphic function on V 1 , regular and non-vanishing at all integer weights k ⩾ − 1 , except possibly at k = r itself, where it is regular.

Proof

It follows easily from the reciprocity laws for diagonal cycles that the ratio

Col b f + ⊗ η g ⊗ ω h ⁡ ( κ ⁢ ( f , g , h ) ) / L p g ⁢ ( f , g , h )

is a function of 𝐤 alone, and this ratio does not vanish at any integer k ⩾ − 1 where f k is classical; it is equal to the fudge factor c k defined above using Section A8.3.

Moreover, since L p g ⁢ ( f , g , h ) is well-defined and non-zero along { ℓ } × V 2 × V 3 , we conclude that c ⁢ ( k ) does not have a pole at 𝐤 (although it might have a zero there). ∎

Remark C2.12

In this study, we have only considered the reciprocity law for the 𝑝-adic 𝐿-function where the dominant family is not the one passing through the critical Eisenstein series point, where some subtle complications may arise. We expect to come back to this issue in forthcoming work.

C2.8 Leading terms

If c ⁢ ( r ) ≠ 0 , then we have thus constructed a class in the cohomology of V ⁢ ( g × h × ψ ) ∗ whose regulator agrees with that of Beilinson–Flach’s Euler system for g × h × ψ .

We claim that if c ⁢ ( r ) = 0 , then in fact κ ⁢ ( f , g , h ) is divisible by 𝑋. If c ⁢ ( r ) = 0 , then κ ⁢ ( E ℓ + 2 crit , g , h ) is in the Selmer group with local condition F + ⁢ V ⁢ ( g ) ∗ ⊗ ̂ V ⁢ ( h ) ∗ , which is zero following the proof of Proposition C2.2 (considering now only one of the 𝑝-adic 𝐿-functions, and therefore a slightly different local condition). So κ ⁢ ( f , g , h ) mod X would have to land in the cohomology of V ⁢ ( g ) sub ∗ ⊗ V ⁢ ( h ) ∗ instead, but then we are seeing the projection into F − , not F + , so by the local properties of Beilinson–Flach elements again (for g × h × τ , instead of g × h × ψ , this time), this is zero as well.

So we can divide out a factor of 𝑋 from both κ ⁢ ( f , g , h ) and c ⁢ ( k ) , and repeat the argument. Since 𝑐 is not identically 0, this must terminate after finitely many steps.

Proposition C2.13

Suppose that n ⩾ 0 is the order of vanishing of c f at k = r . Then X − n ⁢ κ ̂ ⁢ ( f , g , h ) is well-defined and non-zero modulo 𝑋. This leading term projects non-trivially into the quotient H 1 ⁢ ( Q , V ⁢ ( g ) ∗ ⊗ V ⁢ ( h ) ∗ ⁢ ( ψ − 1 ) ⁢ ( − j ) ) . Its image under the Perrin-Riou regulator is given by

c f ∗ ⁢ ( r ) ⋅ log [ r + 1 ] ⋅ L p g ⁢ ( E r + 2 ⁢ ( ψ , τ ) , g , h ) ,

where c f ∗ ⁢ ( r ) ∈ L × .

We denote the resulting class by κ ̂ ∗ ⁢ ( f β , g , h ) . If n = 0 , we have seen above that this class is divisible by log [ r + 1 ] ⁡ ( r − k 2 + k 3 2 ) ; for n > 0 , this is less obvious, but proceeding as before, it follows from the proof of the next theorem.

Theorem C2.14

Under the big image assumptions of [24, §11], we have

κ ̂ ∗ ⁢ ( f β , g , h ) = ( C ⋅ log [ r + 1 ] ⁡ ( r − k 2 + k 3 2 ) ) ⋅ κ d ⁢ ( g , h × ψ ) ,

for some nonzero constant 𝐶 and where κ d ⁢ ( g , h × ψ ) is the two-variable Beilinson–Flach class indexed by the two weight variables.

Proof

The class obtained from the diagonal cycle lies in

V ⁢ ( g ) ∗ ⊗ V ⁢ ( h × ψ ) ∗ ⁢ ( − 1 − r + k 2 + k 3 2 ) ,

so it lives in the same space as the Beilinson–Flach class for j = 1 + r + k 2 + k 3 2 . Then the 𝑑-factor is actually constant over V 2 × V 3 and its value is

d 2 − d − ( k 2 + k 3 − 2 ⁢ j ) ( ε f ε g ψ 2 ) ( d ) − 1 ) = d 2 − d 2 + r ( ε f ε g ψ 2 ) ( d ) − 1 = d 2 ( 1 − d r τ ψ − 1 ( d ) ) .

Note that the “𝑝-decency” hypothesis implies that τ ⁢ ψ − 1 must be non-trivial if r = 0 , so we can choose 𝑑 such that d 2 ⁢ ( 1 − d r ⁢ τ ⁢ ψ − 1 ⁢ ( d ) ) ≠ 0 . Hence we may take

C = d − 2 ⁢ ( 1 − d r ⁢ τ ⁢ ψ − 1 ⁢ ( d ) ) − 1 ⁢ c f ∗ ⁢ ( r ) .

From the previous proposition, together with the explicit reciprocity law for Beilinson–Flach elements, both of the cohomology classes we are considering have the same image under the regulator, so they are equal by Proposition C2.2. Note that we need to assume the big image assumptions of [24, §11] to assure that the Selmer group with Greenberg condition is one-dimensional. ∎

As before, note that, using instead κ ̂ ∗ ⁢ ( f β , g , h ) , the result takes the simpler form

κ ∗ ⁢ ( f β , g , h ) = C ⋅ κ d ⁢ ( g , h × ψ ) .

C3 Deformation of Heegner points

C3.1 Setup

We consider the Heegner point anticyclotomic Euler system of [20], and keep the notation of Section B4 and Section B5. Let f = E r + 2 ⁢ ( ψ , τ ) stand for the Eisenstein series of weight r + 2 and characters ( ψ , τ ) , with ψ ⁢ τ = 1 . As before, let f β be its critical-slope 𝑝-stabilisation. Consider the unique Coleman family 𝐟 passing through f β over some affinoid disc V 1 . We continue assuming that, for all integers k ∈ V 1 ∩ Z ⩾ 0 with k ≠ r , f k is a non-critical-slope cusp form. Recall for this section the module V ac ⁢ ( f ) ∗ defined in Section B5, and consider in the same way V c , ac ⁢ ( f ) ∗ , replacing V ⁢ ( f ) ∗ by V c ⁢ ( f ) ∗ .

C3.2 Families over punctured discs

Proposition C3.1

The cohomology

H 1 ⁢ ( K , V ac ⁢ ( f ) ∗ )

is finitely generated over O ⁢ ( V 1 × W ) , and 𝑋-torsion-free, where X ∈ O ⁢ ( V 1 ) is a uniformiser at 𝑟.

Proof

This follows again by the same argument as Proposition C1.3. ∎

Theorem C3.2

There exists a cohomology class

κ ⁢ ( f , K ) ∈ H 1 ⁢ ( K , 1 X ⁢ V c , ac ⁢ ( f ) ∗ ) ,

with the following interpolation property: if ( k , j ) are integers at least 0 with k ≠ r , then we have

κ ⁢ ( f , K ) ⁢ ( k , j ) = z f k , r ∈ H 1 ⁢ ( K , V ⁢ ( f k ) ∗ ⊗ σ k − j ⁢ σ ̄ j ) ,

where the element z f k , r is as defined in [20, Theorem 5.3.1].

Proof

This follows from the construction of [20], with the usual changes to take into account what happens in a neighbourhood of a critical-slope Eisenstein point. ∎

C3.3 Local properties at 𝑝

Recall that the choice of the embedding K ↪ Q p ̄ singles out one of the primes above 𝑝, that we have called 𝔭. The following result gives information about the vanishing of the local class at 𝔭.

Proposition C3.3

The image of loc p ⁡ ( κ ⁢ ( f , K ) ) in H 1 ⁢ ( K p , 1 X ⁢ F − ⁢ D c ⁢ ( f ) ∗ ) is zero.

Proof

This follows from the fact that the Iwasawa cohomology is torsion-free, and the specialisations away from X = 0 have the required vanishing property. ∎

C3.4 Leading terms at X = 0

Proposition C3.4

The image of κ ⁢ ( f , K ) in the cohomology of the quotient

1 X ⁢ V c , ac ⁢ ( f ) ∗ V ac ⁢ ( f ) ∗ ≅ K p ⁢ ( τ − 1 ) ⁢ ( 1 + r ) ⊗ H Γ ac ⁢ ( − j )

is zero.

Proof

This follows from the local properties of Heegner points [20, Proposition 6.3.2] (in particular, the fact that loc p ⁡ κ ⁢ ( f , K ) factors through the anticyclotomic Iwasawa cohomology of the rank 1 submodule F p + ⁢ D ⁢ ( f ) ∗ ⊂ D ⁢ ( f ) ∗ , as discussed in Section B5). ∎

Note that this result in previous sections relied on the vanishing of particular Selmer groups (and the fact that certain Greenberg conditions were too strong). In this case, this is automatic and does not require any reciprocity law nor any result from the theory of 𝑝-adic 𝐿-functions.

Corollary C3.5

The class κ ⁢ ( f , K ) lifts (uniquely) to H 1 ⁢ ( K , V ac ⁢ ( f ) ∗ ) , and thus has a well-defined image in the module

κ ̂ ⁢ ( f β , K ) ∈ H 1 ⁢ ( K , K p ⁢ ( ψ − 1 ) ⊗ H Γ ac ⁢ ( − j ) ) .

Proposition C3.6

The image of κ ̂ ⁢ ( f β , K ) in the above module is divisible by the logarithm distribution log [ r + 1 ] ∈ H Γ ac .

Proof

This follows the same argument of Proposition C1.8, replacing the cyclotomic algebra with the anticyclotomic one.∎

Since the Iwasawa cohomology is torsion-free, there is a unique class

κ ⁢ ( f β , K ) ∈ H 1 ⁢ ( K , K p ⁢ ( ψ − 1 ) ⊗ H Γ ac ⁢ ( − j ) )

such that

κ ̂ ⁢ ( f β , K ) = log [ r + 1 ] ⋅ κ ⁢ ( f β , K ) .

C3.5 The 𝑝-adic 𝐿-function

Recall the anticyclotomic 𝑝-adic 𝐿-function L p BDP ⁢ ( f ) , that was introduced in Section B5 as a function over V 1 × W . For this construction, the interpolation property also works at k = r , and the complex 𝐿-functions factors as

L ⁢ ( E r + 2 ⁢ ( ψ , τ ) / K × χ ac j , 1 ) = L ⁢ ( ψ / K × σ r + 2 + j ⁢ σ ̄ − j , 1 ) ⋅ L ⁢ ( τ / K × σ j + 1 ⁢ σ ̄ − j − r − 1 , 1 ) = L ⁢ ( ψ / K × σ r + 2 + j ⁢ σ ̄ − j , 1 ) 2 .

Note that we have used that L ⁢ ( τ / K × σ j + 1 ⁢ σ ̄ − j − r − 1 , 1 ) = L ⁢ ( ψ / K × σ r + 2 + j ⁢ σ ̄ − j , 1 ) , which follows from the functional equation together with the condition that ψ ⁢ τ = 1 . This automatically gives an equality of 𝑝-adic 𝐿-functions

L p BDP ⁢ ( E r + 2 ⁢ ( ψ , τ ) ) ⁢ ( χ ac j ) = L p Katz ⁢ ( ψ ) ⁢ ( σ r + 2 + j ⁢ σ ̄ − j ) .

(Alternatively, it directly follows from the construction of [9] that both functions agree.)

C3.6 Perrin-Riou maps

We want to relate the 𝑝-adic 𝐿-function L p BDP ⁢ ( f ) to the image of loc p ⁡ ( κ ⁢ ( f , K ) ) . This factors through the natural map

H 1 ⁢ ( K p , D ac ⁢ ( f ) ∗ ) → H 1 ⁢ ( K p , F + ⁢ D ac ⁢ ( f ) ∗ ) .

Perrin-Riou’s regulator gives a map

Col b f + = ⟨ L F + ⁢ V ⁢ ( f ) ∗ PR ⁢ ( − ) , b f + ⟩ : H 1 ⁢ ( K p , F + ⁢ D ac ⁢ ( f ) ∗ ) → O ⁢ ( V 1 × W )

which interpolates the Perrin-Riou regulators for f k . More precisely, for

z ∈ ( F + ⁢ D ac ⁢ ( f ) ∗ ) φ − 1 = 1 ,

this map sends 𝑧 to

⟨ ι ⁢ ( ( 1 − φ ) ⁢ z ) , b f + ⟩ ,

where 𝜄 is the inclusion

( F + ⁢ D ac ⁢ ( f ) ∗ ) φ − 1 = 0 ↪ ( F + ⁢ D ac ⁢ ( f ) ∗ ⁢ [ 1 / t ] ) φ − 1 = 0 = D cris ⁢ ( F + ⁢ D ac ⁢ ( f ) ∗ ⁢ ( − 1 − k ) ) ⊗ H Γ ac .

Since multiplication by t r + 1 corresponds to multiplication by log [ r + 1 ] on the H Γ ac side, we conclude that

Col b f + ⁡ ( κ ⁢ ( f , K ) ) mod X = c ⁢ ( k ) ⁢ ⟨ L ψ − 1 PR ⁢ ( κ ⁢ ( f , K ) ) , η f k α ⟩ .

Theorem C3.7

We have

Col b f + ⁡ ( κ ⁢ ( f , K ) ) = c ⁢ ( k ) ⋅ L p BDP ⁢ ( f ) ,

where c ⁢ ( k ) is a meromorphic function on V 1 alone, regular and non-vanishing at all integer weights k ⩾ − 1 , except possibly at k = r itself, where it is regular.

Proof

It follows from the reciprocity laws for Heegner points that

Col b f + ⁡ ( κ ⁢ ( f , K ) ) / L p ⁢ ( f , K )

is a function of 𝐤 alone, and this ratio does not vanish at any integer k ⩾ − 1 where f k is classical; it is equal to the fudge factor c k defined above. Since L p BDP ⁢ ( f ) is well-defined and non-zero along { r } × W , we conclude that c ⁢ ( k ) does not have a pole at 𝐤. ∎

C3.7 Leading terms

If c ⁢ ( r ) ≠ 0 , then we have thus constructed a class in Iwasawa cohomology of V ⁢ ( ψ ) ∗ whose regulator agrees with the Euler system of elliptic units. In fact, if c ⁢ ( r ) = 0 , then κ ⁢ ( f , K ) is divisible by 𝑋, so we can divide out a factor of 𝑋 from both κ ⁢ ( f , K ) and c ⁢ ( k ) , and repeat the argument. Since 𝑐 is not identically 0, this must terminate after finitely many steps.

Proposition C3.8

Let n ⩾ 0 be the order of vanishing of c f at k = r . Then X − n ⁢ κ ̂ ⁢ ( f , K ) is well-defined and non-zero modulo 𝑋, and this leading term projects non-trivially into the quotient H 1 ⁢ ( Q , K p ⁢ ( ψ − 1 ) ⊗ H Γ ac ) . Its image under the Perrin-Riou regulator is given by

c f ∗ ⁢ ( r ) ⋅ log [ r + 1 ] ⋅ L p BDP ⁢ ( E r + 2 ⁢ ( ψ , τ ) ) ,

where c f ∗ ⁢ ( r ) ∈ L × .

We denote the resulting class by κ ̂ ∗ ⁢ ( f β , K ) . If n = 0 , we have seen above that this class is divisible by log [ r + 1 ] ; for n > 0 , this is less obvious, but it follows from the proof of the next theorem.

Theorem C3.9

For some nonzero constant 𝐶, we have

κ ̂ ∗ ⁢ ( f β , K ) = ( C ⋅ log [ r + 1 ] ⋅ ( − 1 ) r 2 + j ) ⋅ κ ⁢ ( ψ , K ) ⁢ ( σ − 1 − r − j ⁢ σ ̄ 1 + j ) ,

where κ ⁢ ( ψ , K ) ⁢ ( σ − 1 − r − j ⁢ σ ̄ 1 + j ) is the specialisation at σ − 1 − r − j ⁢ σ ̄ 1 + j of the system of elliptic units defined in Section B4.

Proof

Take as before C = c f ∗ ⁢ ( r ) . This follows from the previous proposition, together with the explicit reciprocity law for elliptic units, that both of the cohomology classes we are considering have the same image under the regulator, and so they must be equal. ∎

Funding source: European Research Council

Award Identifier / Grant number: 101001051

Funding source: Royal Society

Award Identifier / Grant number: NIF\R1\202208

Funding statement: Supported by European Research Council Consolidator Grant No. 101001051 (ShimBSD) (D.L.) and Royal Society Newton International Fellowship NIF\R1\202208 (O.R.).

Acknowledgements

The authors would like to thank Massimo Bertolini, Kazîm Büyükboduk, Marco Seveso, Rodolfo Venerucci, and Sarah Zerbes for informative conversations related to this work, and Victor Rotger for his feedback on an earlier draft. We also thank the anonymous referees for a very careful reading of the text, whose comments notably contributed to improve the exposition of the article.

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Received: 2022-02-08
Revised: 2024-01-12
Published Online: 2024-08-21
Published in Print: 2024-09-01

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Frontmatter
  2. Inverse mean curvature flow and Ricci-pinched three-manifolds
  3. Irregular loci in the Emerton–Gee stack for GL2
  4. Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties
  5. The Lichnerowicz Laplacian on normal homogeneous spaces
  6. Sharp pinching theorems for complete submanifolds in the sphere
  7. On Vafa–Witten equations over Kähler manifolds
  8. Sheaf quantization from exact WKB analysis
  9. Reduced resonance schemes and Chen ranks
  10. Eisenstein degeneration of Euler systems
  11. Erratum to Discriminants and Artin conductors (J. reine angew. Math. 712 (2016), 107–121)
https://doi.org/10.1515/crelle-2024-0057
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Articles in the same Issue

  1. Frontmatter
  2. Inverse mean curvature flow and Ricci-pinched three-manifolds
  3. Irregular loci in the Emerton–Gee stack for GL2
  4. Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties
  5. The Lichnerowicz Laplacian on normal homogeneous spaces
  6. Sharp pinching theorems for complete submanifolds in the sphere
  7. On Vafa–Witten equations over Kähler manifolds
  8. Sheaf quantization from exact WKB analysis
  9. Reduced resonance schemes and Chen ranks
  10. Eisenstein degeneration of Euler systems
  11. Erratum to Discriminants and Artin conductors (J. reine angew. Math. 712 (2016), 107–121)
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