Non-vanishing mod p of theta lifts for (O2n+1, Mp4n
               )

Xiaoyu Zhang

doi:10.1515/forum-2024-0507

Artikel

Non-vanishing mod p of theta lifts for (O_2n+1, Mp_4n)

Xiaoyu Zhang

Veröffentlicht/Copyright: 1. August 2025

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Abstract

We establish the non-vanishing mod p of global theta lifts from an odd definite orthogonal group O 2 ⁢ n + 1 over ℚ to a metaplectic group Mp 4 ⁢ n over ℚ under mild conditions. The problem is closely related to non-vanishing modulo p of toric integrals on O 2 ⁢ n + 1 . For this, we exploit the distribution properties of toric orbits of unipotent elements on O 2 ⁢ n + 1 ⁢ ( ℚ ℓ ) using Ratner’s theorems on unipotent flows and we deduce that the toric integral of a p-primitive automorphic form on O 2 ⁢ n + 1 is non-zero modulo p for infinitely many characters.

Keywords: theta correspondence; ergodic theory

MSC 2020: 11F06; 11F27; 11F46

Communicated by Jan Bruinier

A Toric orbits of unipotent elements

A.1 Basic set-up

Throughout this appendix we fix a prime number ℓ . Write U 2 for the subgroup of SL 2 consisting of unipotent upper triangular matrices

U 2 = { ( 1 b 0 1 ) ∈ SL 2 } .

We fix a connected linear algebraic group 𝐇 over ℚ and a maximal torus 𝐓 (over ℚ ) of 𝐇 , both of which are split at ℓ . We assume that the derived subgroup 𝐇 1 is simply connected and that 𝐇 1 ⁢ ( ℚ ) is discrete in 𝐇 1 ⁢ ( 𝔸 f ) . We assume moreover that there are n algebraic subgroups 𝐇 1 , … , 𝐇 n of 𝐇 such that the following conditions are satisfied:

(A3).

Each 𝐇 j is split at ℓ , isomorphic to either 𝐁 ⁢ [ j ] × for some quaternion algebra 𝐁 ⁢ [ j ] over ℚ or is a unitary group 𝐔 ⁢ [ j ] of rank 2 over an imaginary quadratic field. We fix once and for all an isomorphism

ι j : GL 2 ⁢ ( ℚ ℓ ) ≃ 𝐇 j ⁢ ( ℚ ℓ ) .

Then these subgroups 𝐇 j 1 ⁢ ( ℚ ℓ ) ( j = 1 , … , n ) generate the whole 𝐇 1 ⁢ ( ℚ ℓ ) .
The unipotent subgroups 𝐔 j := ι j ⁢ ( U 2 ⁢ ( ℚ ℓ ) ) all commute with each other. Moreover, 𝐓 ⁢ ( ℚ ℓ ) normalizes 𝐇 j ⁢ ( ℚ ℓ ) and 𝐔 j for j = 1 , … , n and 𝐓 ⁢ ( ℚ ℓ ) acts by conjugation on 𝐔 j via an algebraic character χ j and that these characters χ j are linearly independent in the character group X ∗ ⁢ ( 𝐓 / ℚ ) of 𝐓 / ℚ .

Remark A.1.

Note that the simple-connectedness of 𝐇 1 implies that 𝐇 1 ⁢ ( 𝔸 f ) satisfies the strong approximation property (with respect to the place ℓ ).
By [10, Proposition 1.4], we know that 𝐇 1 ⁢ ( ℚ ) is discrete in 𝐇 1 ⁢ ( 𝔸 f ) if and only if 𝐇 1 ⁢ ( ℚ ) is discrete and co-compact in 𝐇 1 ⁢ ( 𝔸 f ) , if and only if 𝐇 1 ⁢ ( ℝ ) is compact. By assumption, these 𝐇 j 1 ⁢ ( ℝ ) are all compact, 𝐇 j 1 are all simply-connected and 𝐇 j 1 ⁢ ( 𝔸 f ) satisfy the strong approximation property (with respect to the place ℓ ).
The two groups 𝐁 ⁢ [ j ] × and 𝐔 ⁢ [ j ] are closely related as follows: for each quaternion algebra 𝐁 over ℚ with a principal involution τ, we can define a dimension 2 Hermitian space over a quadratic extension K of ℚ (which is a maximal commutative subalgebra of 𝐁 and is stable under the involution τ) and write 𝐒𝐔 for the corresponding special unitary group associated to this space. Write 𝐁 1 for the set of elements of 𝐁 whose reduced norm is 1. Then 𝐁 1 ≃ 𝐒𝐔 as algebraic groups over ℚ . So the groups 𝐇 j can all be seen as extensions of a rank one torus (over ℚ ) by some 𝐁 1 . From (2) we have an isomorphism ∏ j = 1 n 𝐔 j ≃ ℚ ℓ n of ℓ -adic groups. Moreover, 𝐓 ⁢ ( ℚ ℓ ) normalizes each 𝐇 j 1 ⁢ ( ℚ ℓ ) .

The following are two examples of 𝐇 satisfying the above assumptions. We refer to Section A.5 for other more examples.

Examples A.2.

𝐇 is 𝐁 × , where 𝐁 is a definite quaternion algebra over ℚ ,
𝐇 is a unitary group 𝐔 of rank 2 over an imaginary quadratic field.

We assume that 𝐇 is split at ℓ . In this case we fix a maximal torus 𝐓 of 𝐇 which is also split at ℓ . Then we take n = 1 , 𝐇 1 = 𝐇 , 𝐔 1 is a unipotent subgroup of 𝐇 1 ⁢ ( ℚ ℓ ) normalized by 𝐓 ⁢ ( ℚ ℓ ) .

A.2 Ratner’s theorems and commensurability

A.2.1 SL(2) case

In this subsection we recall Ratner’s orbit closure theorem and uniform distribution theorem on unipotent flows. Write 𝐦 to be the normalized Haar measure on ℚ ℓ . For a subset κ of ℚ ℓ and an integer N, we write

κ N = κ ⁢ ℓ - N = { k ⁢ ℓ - N ∣ k ∈ κ } .

Let 𝐇 be as in the preceding section and put 𝐆 = 𝐇 1 ⁢ ( ℚ ℓ ) . A lattice in 𝐆 is a discrete and cocompact subgroup. Then Ratner’s theorems give ([26, Theorems 2 and 3]):

Theorem A.3.

Let Γ be a lattice in G and let U be a subgroup of G generated by one-parameter unipotent subgroups of G .

(Orbit closure theorem) For any x ∈ Γ \ 𝐆 , the closure x ⁢ 𝐔 in Γ \ 𝐆 is of the form x ⁢ 𝐋 for some closed subgroup 𝐋 of 𝐆 containing 𝐔 .
(Uniform distribution theorem) Let 𝐔 = { 𝐮 ⁢ ( t ) | t ∈ ℚ ℓ } be a one-parameter unipotent subgroup of 𝐆 . Write μ 𝐋 for the unique Borel measure on Γ \ 𝐆 invariant under the action of 𝐋 and supported on x ⁢ 𝐋 . Then for any locally constant function f : Γ \ 𝐆 → ℂ and any compact open subset κ ⊂ ℚ ℓ , one has

lim N → + ∞ ⁡ 1 𝐦 ⁢ ( κ N ) ⁢ ∫ κ N f ⁢ ( x ⁢ 𝐮 ⁢ ( t ) ) ⁢ 𝑑 t = ∫ Γ \ 𝐆 f ⁢ 𝑑 μ 𝐋 .

Remark A.4.

In [26], f is assumed to be continuous. However, since Γ \ 𝐆 is compact, f can be uniformly approximated by locally constant functions on Γ \ 𝐆 . Thus the last conclusion in the above theorem is equivalent to the one given in [26].

The proofs of the above results rely on a careful study for the case ([26, Theorem 6])

𝐆 = SL 2 ⁢ ( ℚ ℓ ) r ,

which we will also need in the following. Let Γ 1 , … , Γ r be lattices in SL 2 ⁢ ( ℚ ℓ ) and write

Γ = Γ 1 × ⋯ × Γ r .

We write Δ the diagonal embedding

Δ : SL 2 ⁢ ( ℚ ℓ ) → SL 2 ⁢ ( ℚ ℓ ) r

and V a one-parameter unipotent subgroup of SL 2 ⁢ ( ℚ ℓ )

V = { v ⁢ ( t ) ∣ t ∈ ℚ ℓ } .

So Δ ⁢ ( V ) is a one-parameter unipotent subgroup of SL 2 ⁢ ( ℚ ℓ ) r .

Theorem A.5.

For any g = ( g 1 , … , g r ) ∈ SL 2 ⁢ ( Q ℓ ) r , the closure Γ ⁢ g ⁢ Δ ⁢ ( V ) inside SL 2 ⁢ ( Q ℓ ) r is of the form Γ ⁢ g ⁢ L for a closed subgroup L of SL 2 ⁢ ( Q ℓ ) r containing Δ ⁢ ( V ) and there is an element c ∈ V r such that c ⁢ L ⁢ c - 1 ⊃ Δ ⁢ ( SL 2 ⁢ ( Q ℓ ) ) . Moreover, there is a unique L -invariant Borel measure μ L on Γ \ SL 2 ⁢ ( Q ℓ ) r supported on Γ ⁢ g ⁢ L , and the measure μ L is ergodic for L : for any locally constant function f : Γ \ SL 2 ⁢ ( Q ℓ ) r → C and any compact open subset κ ⊂ Q ℓ ,

lim N → + ∞ ⁡ 1 𝐦 ⁢ ( κ N ) ⁢ ∫ κ N f ⁢ ( g ⁢ v ⁢ ( t ) ) ⁢ 𝑑 t = ∫ Γ \ SL 2 ⁢ ( ℚ ℓ ) r f ⁢ 𝑑 μ 𝐋 .

The part c ⁢ 𝐋 ⁢ c - 1 ⊃ Δ ⁢ ( SL 2 ⁢ ( ℚ ℓ ) ) for some c ∈ V r comes from [26, Theorem 6] and [27, Theorem 1.1]. We can say something more for the closed subgroup 𝐋 under certain conditions. For this we need the notion of V-commensurability: we say two lattices Γ , Γ ′ of SL 2 ⁢ ( ℚ ℓ ) are V - commensurable, if there is an element v ∈ V such that Γ and v ⁢ Γ ′ ⁢ v - 1 are commensurable (that is, Γ ∩ v ⁢ Γ ′ ⁢ v - 1 has finite index in both Γ and v ⁢ Γ ′ ⁢ v - 1 ). Then one has ([4, Proposition 2.35]):

Theorem A.6.

Maintain the notations of the preceding theorem. If for any i ≠ j ∈ { 1 , … , r } , the lattices g i - 1 ⁢ Γ i ⁢ g i and g j - 1 ⁢ Γ j ⁢ g j are not V-commensurable, then L = SL 2 ⁢ ( Q ℓ ) r .

Remark A.7.

In fact, the converse is also true (see loc.cit). But we will not need this in the following.

A.2.2 Adelic reformulation

For later applications, it is useful to give an adelic point of view of the preceding result. Let 𝐇 be as in Examples A.2.

Fix a compact open subgroup K ℓ of 𝐇 ⁢ ( 𝔸 f ℓ ) and let 𝐇 1 ⁢ ( ℚ ℓ ) act on the right on 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( 𝔸 f ) / K ℓ . For an element g ∈ 𝐇 ⁢ ( 𝔸 f ) , write Γ K ℓ ⁢ ( g ) for the stabilizer inside 𝐇 1 ⁢ ( ℚ ℓ ) of the double coset 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ K ℓ ∈ 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( 𝔸 f ) / K ℓ .

Lemma A.8.

The stabilizer Γ K ℓ ⁢ ( g ) is a lattice of H 1 ⁢ ( Q ℓ ) .

Proof.

Since 𝐇 1 ⊂ 𝐇 are linear algebraic groups over ℚ , we have

𝐇 ⁢ ( ℚ ) ∩ 𝐇 1 ⁢ ( ℚ ℓ ) = 𝐇 1 ⁢ ( ℚ ) .

Write Γ to be the image of the projection map 𝐇 1 ⁢ ( 𝔸 f ) → 𝐇 1 ⁢ ( ℚ ℓ ) of the following subgroup:

𝐇 ⁢ ( ℚ ) ∩ ( g ⁢ K ℓ ⁢ g - 1 × 𝐇 1 ⁢ ( ℚ ℓ ) ) = 𝐇 1 ⁢ ( ℚ ) ∩ ( g ⁢ K ℓ ⁢ g - 1 × 𝐇 1 ⁢ ( ℚ ℓ ) ) ⊂ 𝐇 1 ⁢ ( 𝔸 f ) .

Then it is easy to see that

Γ K ℓ ⁢ ( g ) = g ℓ - 1 ⁢ Γ ⁢ g ℓ .

Write W = g ⁢ K ℓ ⁢ g - 1 × 𝐇 1 ⁢ ( ℚ ℓ ) ⊂ 𝐇 1 ⁢ ( 𝔸 f ) . Then the continuous injective map

( 𝐇 1 ⁢ ( ℚ ) ∩ W ) \ W → 𝐇 1 ⁢ ( ℚ ) \ 𝐇 1 ⁢ ( 𝔸 f )

is open since W is open in 𝐇 1 ⁢ ( 𝔸 f ) and is also surjective since 𝐇 1 ⁢ ( 𝔸 f ) satisfies the strong approximation property. Thus it is a homeomorphism. By assumption 𝐇 1 ⁢ ( ℚ ) is a lattice inside 𝐇 1 ⁢ ( 𝔸 f ) . This implies that 𝐇 1 ⁢ ( ℚ ) ∩ W is a lattice in W and since the factor g ⁢ K ℓ ⁢ g - 1 is compact, Γ is also a lattice in 𝐇 1 ⁢ ( ℚ ℓ ) by [30, p. 105, Lemme 1.2]. Thus Γ K ℓ ⁢ ( g ) is a lattice in 𝐇 1 ⁢ ( ℚ ℓ ) . ∎

Since 𝐇 1 ⁢ ( 𝔸 f ) satisfies strong approximation property with respect to ℓ , one has

g - 1 ⁢ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ 𝐇 1 ⁢ ( ℚ ℓ ) ⁢ K ℓ = g - 1 ⁢ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ .

One deduces that the following natural map is a homeomorphism of topological spaces:

Γ K ℓ ⁢ ( g ) \ 𝐇 1 ⁢ ( ℚ ℓ ) ≃ 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ / K ℓ , Γ K ℓ ⁢ ( g ) ⁢ h ↦ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ h ⁢ K ℓ = 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ K ℓ ⁢ h .

Note that this homeomorphism is 𝐇 1 ⁢ ( ℚ ℓ ) -equivariant. Thus the measure μ 𝐋 on Γ K ℓ ⁢ ( g ) \ 𝐇 1 ⁢ ( ℚ ℓ ) as in Theorem A.3 corresponds to a measure on the right-hand side which is 𝐋 -invariant. We denote this measure by μ g , V . So Theorem A.3 gives:

Corollary A.9.

For any locally constant function f : H ⁢ ( Q ) \ H ⁢ ( A f ) / K ℓ → C and any compact open subset κ of Q ℓ ,

lim N → + ∞ ⁡ 1 𝐦 ⁢ ( κ N ) ⁢ ∫ κ N f ⁢ ( g ⁢ v ⁢ ( t ) ) ⁢ 𝑑 t = ∫ 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ / K ℓ f ⁢ 𝑑 μ g , V .

We want to put some conditions on g to ensure that 𝐋 is as large as possible, as in Theorem A.6. We proceed this in two steps in the next two subsections.

A.2.3 Single-copy case

We now consider a higher-dimensional generalization of the above results. In this subsection we treat the case r = 1 .

Let 𝐇 be as in Section A.1. We fix isomorphisms

u j : ℚ ℓ ≃ 𝐔 j , t ↦ u j ⁢ ( t ) for all ⁢ j = 1 , … , n .

Let g ∈ 𝐓 ⁢ ( 𝔸 f ) and K ℓ be a compact open subgroup of 𝐇 ⁢ ( 𝔸 f ℓ ) . We have:

Lemma A.10.

The stabilizer Γ K ℓ ⁢ ( g ) is a lattice in H 1 ⁢ ( Q ℓ ) . Similarly, the intersection

Γ K ℓ ( j ) ⁢ ( g ) := Γ K ℓ ⁢ ( g ) ∩ 𝐇 j 1 ⁢ ( ℚ ℓ )

is a lattice in H j 1 ⁢ ( Q ℓ ) .

Proof.

Write Γ for the image under the projection map 𝐇 1 ⁢ ( 𝔸 f ) → 𝐇 1 ⁢ ( ℚ ℓ ) of the following subgroup:

𝐇 ⁢ ( ℚ ) ∩ ( g ⁢ K ℓ ⁢ g - 1 × 𝐇 1 ⁢ ( ℚ ℓ ) ) ⊂ 𝐇 1 ⁢ ( 𝔸 f ) .

Then we have

Γ K ℓ ⁢ ( g ) = g ℓ - 1 ⁢ Γ ⁢ g ℓ .

Write Γ j = Γ ∩ 𝐇 j 1 ⁢ ( ℚ ℓ ) for the projection image to 𝐇 j 1 ⁢ ( ℚ ℓ ) of the following subgroup:

𝐇 j ⁢ ( ℚ ) ∩ ( g ⁢ K ℓ ⁢ g - 1 × 𝐇 j 1 ⁢ ( ℚ ℓ ) ) ⊂ 𝐇 j 1 ⁢ ( 𝔸 f ) .

Since g ℓ ∈ 𝐓 ⁢ ( ℚ ℓ ) normalizes 𝐇 j ⁢ ( ℚ ℓ ) and 𝐇 j 1 ⁢ ( ℚ ℓ ) , we have

Γ K ℓ ( j ) ⁢ ( g ) = g ℓ - 1 ⁢ Γ j ⁢ g ℓ .

Now the same proof as for Lemma A.8 shows that Γ, resp. Γ j is a lattice inside 𝐇 1 ⁢ ( ℚ ℓ ) , resp. 𝐇 j 1 ⁢ ( ℚ ℓ ) (using the fact that 𝐇 1 ⁢ ( 𝔸 f ) , resp. 𝐇 j 1 ⁢ ( 𝔸 f ) satisfies the strong approximation property and 𝐇 1 ⁢ ( ℚ ) , resp. 𝐇 j 1 ⁢ ( ℚ ) is a lattice in 𝐇 1 ⁢ ( 𝔸 f ) , resp. 𝐇 j 1 ⁢ ( 𝔸 f ) ). ∎

For a lattice Γ ′ in 𝐇 j 1 ⁢ ( ℚ ℓ ) , we define the commensurator of Γ ′ inside 𝐇 j ⁢ ( ℚ ℓ ) to be

𝒞 j ⁢ ( Γ ′ ) := { h ∈ 𝐇 j ⁢ ( ℚ ℓ ) ∣ Γ ′ ⁢ and ⁢ h ⁢ Γ ′ ⁢ h - 1 ⁢ are commensurable } .

Remark A.11.

By [4, Lemma 2.19], we know

𝒞 j ⁢ ( Γ K ℓ ( j ) ⁢ ( g ) ) = 𝐇 j ⁢ ( ℚ ) ⁢ Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) , j = 1 , … , n .

Since all compact open subgroups K ℓ of 𝐇 1 ⁢ ( 𝔸 f ℓ ) are commensurable, 𝒞 j ⁢ ( Γ K ℓ ( j ) ⁢ ( g ) ) does not depend on K ℓ .

For a compact open subset κ of ℚ ℓ , an n-tuple of integers N ¯ = ( N 1 , N 2 , … , N n ) ∈ ℕ n and an element t ¯ = ( t 1 , … , t n ) ∈ ℚ ℓ n , write

κ N ¯ = κ N 1 × ⋯ × κ N n ⊂ ℚ ℓ n ,

𝐦 ⁢ ( κ N ¯ ) = 𝐦 ⁢ ( κ N 1 ) × ⋯ × 𝐦 ⁢ ( κ N n ) ,

u ⁢ ( t ¯ ) = u 1 ⁢ ( t 1 ) × ⋯ × u n ⁢ ( t n ) ∈ ∏ j = 1 n 𝐔 j .

Moreover, we write N ¯ → + ∞ to mean N 1 , N 2 , … , N n → + ∞ . Then:

Theorem A.12.

The subset Γ K ℓ ⁢ ( g ) ⁢ ∏ j = 1 n U j is dense in H 1 ⁢ ( Q ℓ ) . Moreover, if we write μ g for the unique Borel measure on H ⁢ ( Q ) \ H ⁢ ( Q ) ⁢ g ⁢ H 1 ⁢ ( A f ) ⁢ K ℓ / K ℓ invariant under H 1 ⁢ ( Q ℓ ) , then for any locally constant complex-valued function f on H ⁢ ( Q ) \ H ⁢ ( A f ) / K ℓ and any compact open subset κ of Q ℓ , we have the following equidistribution result:

(A.1) lim N ¯ → + ∞ ⁡ 1 𝐦 ⁢ ( κ N ¯ ) ⁢ ∫ κ N ¯ f ⁢ ( g ⁢ u ⁢ ( t ¯ ) ) ⁢ 𝑑 t ¯ = ∫ 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ / K ℓ f ⁢ 𝑑 μ g .

Proof.

To ease notations, we write

X = 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( ℚ ) ⁢ g ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ / K ℓ .

For the density, by Theorem A.3, the closure of Γ K ℓ ⁢ ( g ) ⁢ ∏ j 𝐔 j is of the form Γ K ℓ ⁢ ( g ) ⁢ 𝐋 for a closed subgroup 𝐋 of 𝐇 1 ⁢ ( ℚ ℓ ) containing ∏ j 𝐔 j . So for any h j ∈ 𝐇 j 1 ⁢ ( ℚ ℓ ) and any open subset W j of 𝐇 j 1 ⁢ ( ℚ ℓ ) containing h j , we have

( Γ K ℓ ( j ) ⁢ ( g ) ⁢ 𝐔 j ) ∩ ( Γ K ℓ ( j ) ⁢ ( g ) ⁢ h j ⁢ W j ) ≠ ∅ .

Thus for any open subset W of 𝐇 1 ⁢ ( ℚ ℓ ) containing h j , the intersection W ∩ 𝐇 j 1 ⁢ ( ℚ ℓ ) is open in 𝐇 j 1 ⁢ ( ℚ ℓ ) for any j = 1 , … , n and one deduces from the previous non-empty intersection

( Γ K ℓ ⁢ ( g ) ⁢ ∏ j ′ 𝐔 j ′ ) ∩ ( Γ K ℓ ⁢ ( g ) ⁢ h j ⁢ W ) ≠ ∅ .

Therefore 𝐋 contains all these 𝐇 j 1 ⁢ ( ℚ ℓ ) , which generate the whole 𝐇 1 ⁢ ( ℚ ℓ ) , thus 𝐋 = 𝐇 1 ⁢ ( ℚ ℓ ) .

One deduces that the measure μ 𝐋 is the unique Borel measure on Γ K ℓ ⁢ ( g ) \ 𝐇 1 ⁢ ( ℚ ℓ ) which is 𝐇 1 ⁢ ( ℚ ℓ ) -invariant. This measure is transferred to the unique Borel measure on X, via transport de structure by the following natural 𝐇 1 ⁢ ( ℚ ℓ ) -equivariant homeomorphism:

Γ K ℓ ⁢ ( g ) \ 𝐇 1 ⁢ ( ℚ ℓ ) ≃ X .

This is the measure μ g in the theorem.

To prove (A.1), we argue as follows: for any N ¯ as above, we define a Borel probability measure μ N ¯ on X by the formula

∫ X f ⁢ ( x ) ⁢ 𝑑 μ N ¯ ⁢ ( x ) = 1 𝐦 ⁢ ( κ N ¯ ) ⁢ ∫ t ¯ ∈ κ N ¯ f ⁢ ( u ⁢ ( t ¯ ) ) ⁢ 𝑑 t ¯ for any continuous/locally constant ⁢ f : X → ℂ .

As X is compact, for any sequence N ¯ ( k ) = ( N 1 , k , … , N n , k ) with N i , k → + ∞ for k → + ∞ , there is a subsequence { N ¯ ( k s ) } s ∈ ℕ such that μ N ¯ ( k s ) converges (under the weak topology) to a Borel probability measure μ ′ on X. We claim that μ ′ is 𝐇 1 ⁢ ( ℚ ℓ ) -invariant and thus we necessarily have μ ′ = μ g , which finishes the proof of the theorem.

To prove the claim, note that ∏ j = 1 n 𝐔 j preserves μ ′ (because κ is a compact open subset of ℚ ℓ ). For h ∈ 𝐇 1 ⁢ ( ℚ ℓ ) , write μ ′ ∘ h for the right translation of h on μ ′ , that is,

∫ X f ⁢ ( x ) ⁢ d ⁢ ( μ ′ ∘ h ) ⁢ ( x ) := ∫ X f ⁢ ( x ⁢ h ) ⁢ 𝑑 μ ′ ⁢ ( x ) .

Any locally constant function f : X → ℂ is invariant under the right translation by a compact open subgroup K ′ of 𝐇 1 ⁢ ( ℚ ℓ ) . Thus we have

∫ X f ⁢ ( x ) ⁢ d ⁢ ( μ ′ ∘ k ) ⁢ ( x ) = ∫ X f ⁢ ( x ) ⁢ 𝑑 μ ′ ⁢ ( x ) for all ⁢ k ∈ K ′ .

Fix j = 1 , … , n , since 𝐇 j 1 ⁢ ( Q ℓ ) ∩ K ′ is an open subgroup in 𝐇 j 1 ⁢ ( ℚ ℓ ) , 𝐔 j and 𝐇 j 1 ⁢ ( ℚ ℓ ) ∩ K ′ generate 𝐇 j 1 ⁢ ( ℚ ℓ ) , then one deduces

∫ X f ⁢ ( x ) ⁢ d ⁢ ( μ ′ ∘ h j ) ⁢ ( x ) = ∫ X f ⁢ ( x ) ⁢ 𝑑 μ ′ ⁢ ( x ) for all ⁢ h j ∈ 𝐇 j 1 ⁢ ( ℚ ℓ ) .

Since this is true for arbitrary locally constant function f, we deduce that μ ′ is invariant under 𝐇 j 1 ⁢ ( ℚ ℓ ) for any j, and therefore also invariant under 𝐇 1 ⁢ ( ℚ ℓ ) (it is generated by these 𝐇 j 1 ⁢ ( ℚ ℓ ) ). This proves our claim. ∎

A.2.4 Multi-copy case

In this subsection we treat the case r > 1 .

Again let 𝐇 be as in the preceding section, let K ℓ be a compact open subgroup of 𝐇 ⁢ ( 𝔸 f ℓ ) and g = ( g 1 , … , g r ) with g 1 , … , g r ∈ 𝐓 ⁢ ( 𝔸 f ) such that their ℓ -th components satisfy

(A.2) ( g k ) ℓ ⁢ ( g i ) ℓ - 1 ∉ 𝐓 ⁢ ( ℚ ℓ ) ∩ ( ⋃ j = 1 n 𝐇 j ⁢ ( ℚ ) ⁢ Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) ) for all ⁢ k ≠ i ∈ { 1 , … , r } .

As in the preceding subsection, Γ K ℓ ( j ) ⁢ ( g i ) is a lattice inside 𝐇 j 1 ⁢ ( ℚ ℓ ) . We write

Γ K ℓ ( j ) ⁢ ( g ) = Γ K ℓ ( j ) ⁢ ( g 1 ) × ⋯ × Γ K ℓ ( j ) ⁢ ( g r ) , Γ K ℓ ⁢ ( g ) = Γ K ℓ ⁢ ( g 1 ) × ⋯ × Γ K ℓ ⁢ ( g r ) ,

which is a lattice in 𝐇 j 1 ⁢ ( ℚ ℓ ) r , resp. 𝐇 1 ⁢ ( ℚ ℓ ) r . Then we have:

Lemma A.13.

Fix an element h ∈ T ⁢ ( A f ) . For any j = 1 , … , n , Γ K ℓ ( j ) ⁢ ( Δ ⁢ ( h ) ⁢ g ) ⁢ Δ ⁢ ( U j ) is dense in H j 1 ⁢ ( Q ℓ ) r . Similarly, Γ K ℓ ⁢ ( Δ ⁢ ( h ) ⁢ g ) ⁢ Δ ⁢ ( ∏ j = 1 n U j ) is dense in H 1 ⁢ ( Q ℓ ) r .

Proof.

Fix i ≠ k ∈ { 1 , … , r } . We claim there is no w ∈ 𝐔 j such that Γ K ℓ ( j ) ⁢ ( h ⁢ g i ) and w ⁢ Γ K ℓ ( j ) ⁢ ( h ⁢ g k ) ⁢ w - 1 are commensurable: otherwise, recall the expression for Γ K ℓ ( j ) ⁢ ( h ⁢ g i ) , using Remark A.11, we have the following:

( h ⁢ g i ) ℓ - 1 ⁢ 𝒞 j ⁢ ( Γ K ℓ ( j ) ⁢ ( 1 ) ) ⁢ ( h ⁢ g i ) ℓ = 𝒞 j ⁢ ( Γ K ℓ ( j ) ⁢ ( h ⁢ g i ) ) = 𝒞 j ⁢ ( w ⁢ Γ K ℓ ( j ) ⁢ ( h ⁢ g k ) ⁢ w - 1 )

= w - 1 ⁢ 𝒞 j ⁢ ( Γ K ℓ ( j ) ⁢ ( h ⁢ g k ) ) ⁢ w

= ( h ℓ ⁢ ( g k ) ℓ ⁢ w ) - 1 ⁢ 𝒞 j ⁢ ( Γ K ℓ ( j ) ⁢ ( 1 ) ) ⁢ ( h ℓ ⁢ ( g k ) ℓ ⁢ w ) .

We write b = h ℓ ⁢ ( g k ) ℓ ⁢ w ⁢ ( g i ) ℓ - 1 ⁢ h ℓ - 1 . Then one deduces

b ∈ 𝐇 j ⁢ ( ℚ ) ⁢ Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) .

Note that h ℓ , ( g i ) ℓ , ( g k ) ℓ normalize 𝐔 j , so there is another w ′ ∈ 𝐔 j such that b = ( g k ) ℓ ⁢ ( g i ) ℓ - 1 ⁢ w ′ . Recall 𝐓 ⁢ ( ℚ ℓ ) normalizes 𝐇 j ⁢ ( ℚ ℓ ) , thus for any s ∈ 𝐓 ⁢ ( ℚ ℓ ) and z ∈ Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) , s ⁢ z ⁢ s - 1 ∈ Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) . So we have a continuous morphism of topological groups

φ : 𝐓 ⁢ ( ℚ ℓ ) → Auto ⁢ ( Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) )

induced by the conjugate action. Thus for any s ∈ Ker ⁢ ( φ ) ∩ 𝐓 ⁢ ( ℚ ) (which is dense in Ker ⁢ ( φ ) ), since 𝐓 ⁢ ( ℚ ℓ ) also normalizes 𝐔 j , the commutator s ⁢ b ⁢ s - 1 ⁢ b - 1 ∈ 𝐇 ⁢ ( ℚ ) is a unipotent element in 𝐇 1 ⁢ ( ℚ ℓ ) . Since 𝐇 1 ⁢ ( ℝ ) is compact, 𝐇 1 ⁢ ( ℚ ) cannot contain non-trivial unipotent elements, thus one must have s ⁢ b ⁢ s - 1 ⁢ b - 1 = 1 for any s ∈ Ker ⁢ ( φ ) ∩ 𝐓 ⁢ ( ℚ ) . Moreover, Ker ⁢ ( φ ) ∩ 𝐓 ⁢ ( ℚ ) is dense in Ker ⁢ ( φ ) , so b commutes with all Ker ⁢ ( φ ) , an open subgroup of 𝐓 ⁢ ( ℚ ℓ ) . Thus b commutes with 𝐓 ⁢ ( ℚ ℓ ) . This implies that b ∈ 𝐓 ⁢ ( ℚ ℓ ) since 𝐓 ⁢ ( ℚ ℓ ) is a maximal torus in 𝐇 ⁢ ( ℚ ℓ ) . On the other hand, b = ( g k ) ℓ ⁢ ( g i ) ℓ - 1 ⁢ w ′ . So we must have w ′ = 1 and therefore

( g k ) ℓ ⁢ ( g i ) ℓ - 1 ∈ 𝐓 ⁢ ( ℚ ℓ ) ∩ ( 𝐇 j ⁢ ( ℚ ) ⁢ Z ⁢ ( 𝐇 j ⁢ ( ℚ ℓ ) ) ) .

This contradicts our assumption (A.2) on g i , g k . So Γ K ℓ ( j ) ⁢ ( h ⁢ g i ) and Γ K ℓ ( j ) ⁢ ( h ⁢ g k ) are not 𝐔 j -commensurable for any i ≠ k .

Applying Theorem A.6, we see that Γ K ℓ ( j ) ⁢ ( Δ ⁢ ( h ) ⁢ g ) ⁢ Δ ⁢ ( 𝐔 j ) is dense in 𝐇 j 1 ⁢ ( ℚ ℓ ) r for any j = 1 , … , n . Apply Theorem A.3 (1) and we see that Γ K ℓ ⁢ ( Δ ⁢ ( h ) ⁢ g ) ⁢ Δ ⁢ ( ∏ j 𝐔 j ) is dense in 𝐇 1 ⁢ ( ℚ ℓ ) r since these 𝐇 j 1 ⁢ ( ℚ ℓ ) r generate 𝐇 1 ⁢ ( ℚ ℓ ) r . ∎

Analogue to Theorem A.12, we have:

Theorem A.14.

Fix h ∈ T ⁢ ( A f ) . For any locally constant function f on ( H ⁢ ( Q ) \ H ⁢ ( A f ) / K ℓ ) r and any compact open subset κ of Q ℓ , we have

(A.3) lim N ¯ → + ∞ ⁡ 1 𝐦 ⁢ ( κ N ¯ ) ⁢ ∫ κ N ¯ f ⁢ ( Δ ⁢ ( h ) ⁢ g ⁢ Δ ⁢ ( u ⁢ ( t ¯ ) ) ) ⁢ 𝑑 t ¯ = ∫ ∏ i G ⁢ ( ℚ ) \ G ⁢ ( ℚ ) ⁢ h ⁢ g i ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ / K ℓ f ⁢ 𝑑 μ Δ ⁢ ( h ) ⁢ g .

Here μ Δ ⁢ ( h ) ⁢ g is the product of the measures μ h ⁢ g i on G ⁢ ( Q ) \ G ⁢ ( Q ) ⁢ h ⁢ g i ⁢ H 1 ⁢ ( A f ) ⁢ K ℓ / K ℓ for i = 1 , … , r .

A.3 Main result

Let 𝐇 , 𝐓 , 𝐇 j , 𝐔 j be as in Section A.1. We fix also the following data:

r elements g 1 , … , g r ∈ 𝐓 ⁢ ( 𝔸 f ) such that (A.2) is satisfied. In particular, these g 1 , … , g r are all distinct. We write

g = ( g 1 , g 2 , … , g r ) .
a compact open subgroup K ℓ of 𝐇 ⁢ ( 𝔸 f ℓ ) .
an open subgroup 𝒢 of 𝐓 ⁢ ( ℚ ) \ 𝐓 ⁢ ( 𝔸 f ) and a Haar measure μ 𝒢 on 𝒢 .

Then integrating both sides (A.3) with respect to the variable h ∈ 𝒢 , we get

(A.4) lim N ¯ → + ∞ ⁡ 1 𝐦 ⁢ ( κ N ¯ ) ⁢ ∫ κ N ¯ 𝑑 t ¯ ⁢ ∫ 𝒢 f ⁢ ( Δ ⁢ ( h ) ⁢ g ⁢ Δ ⁢ ( u ⁢ ( t ¯ ) ) ) ⁢ 𝑑 μ 𝒢 ⁢ ( h ) = ∫ 𝒢 𝑑 μ 𝒢 ⁢ ( h ) ⁢ ∫ ∏ i 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( ℚ ) ⁢ h ⁢ g i ⁢ 𝐇 1 ⁢ ( 𝔸 f ) ⁢ K ℓ / K ℓ f ⁢ 𝑑 μ Δ ⁢ ( h ) ⁢ g .

Under condition (A3), we can choose n torus subgroups T 1 ≃ ℚ ℓ × , … , T n ≃ ℚ ℓ × of 𝐓 ⁢ ( ℚ ℓ ) whose conjugate action on ∏ j = 1 n 𝐔 j are pairwise distinct.

Proposition A.15.

Fix a compact open subgroup K ℓ ⊂ H ⁢ ( Q ℓ ) . Then, for any m 1 , … , m n ≫ 0 , we have

𝒢 ⁢ ∏ i = 1 n u i ⁢ ( ( 1 + ℓ m i ⁢ ℤ ℓ ) ⁢ ℓ - k i ) ⁢ K ℓ = 𝒢 ⁢ ∏ i = 1 n u i ⁢ ( ℓ - k i ) ⁢ K ℓ for all ⁢ k 1 , … , k n ≥ 0 .

Proof.

Recall we fixed isomorphisms

σ i : T i ≃ ℚ ℓ × , u j : ℚ ℓ ≃ 𝐔 j .

Thus there is an integer r i , j such that the conjugate action φ of T i on 𝐔 j is given by

φ ⁢ ( σ i ⁢ ( x i ) , u j ⁢ ( t ) ) = u j ⁢ ( x i r i , j ⁢ t ) for all ⁢ x i ∈ ℚ ℓ × , t ∈ ℚ ℓ .

Under condition (A3), the matrix ( r i , j ) i , j = 1 n is non-singular. For any x i ∈ ℚ ℓ × viewed as an element in T i and any t i ∈ ℚ ℓ , one has

∏ j = 1 σ j ⁢ ( x j ) ⁢ ∏ j = 1 u j ⁢ ( t j ⁢ ℓ - k j ) ⁢ ∏ j = 1 σ j ⁢ ( x j ) - 1 = ∏ j = 1 u j ⁢ ( ∏ i = 1 n x i r i , j ⁢ t j ⁢ ℓ - k j ) .

For any m 1 , … , m n ≫ 0 , we can choose m 1 ′ , … , m n ′ > 0 such that the system of equations on the variables x 1 , … , x n

{ x 1 r 1 , 1 ⁢ ⋯ ⁢ x n r n , 1 = t 1 - 1 ∈ 1 + ℓ m 1 ⁢ ℤ ℓ , x 1 r 1 , 2 ⁢ ⋯ ⁢ x n r n , 2 = t 2 - 1 ∈ 1 + ℓ m 2 ⁢ ℤ ℓ , ⋮ x 1 r 1 , n ⁢ ⋯ ⁢ x n r n , n = t n - 1 ∈ 1 + ℓ m n ⁢ ℤ ℓ

always has a solution in x i ∈ 1 + ℓ m i ′ ⁢ ℤ ℓ ( i = 1 , … , n ). We write T i ⁢ ( m i ′ ) := σ i ⁢ ( 1 + ℓ m i ′ ⁢ ℤ ℓ ) . Then we have the following identity:

∏ i = 1 n T i ⁢ ( m i ′ ) ⁢ ∏ i = 1 n u i ⁢ ( t i ⁢ ℓ - k i ) ⁢ ∏ i = 1 n T i ⁢ ( m i ′ ) = ∏ i = 1 n T i ⁢ ( m i ′ ) ⁢ ∏ i = 1 n u i ⁢ ( ℓ - k i ) ⁢ ∏ i = 1 n T i ⁢ ( m i ′ ) for all ⁢ t i ∈ 1 + ℓ m i ⁢ ℤ ℓ .

Moreover, for m 1 , … , m n ≫ 0 , we can choose m 1 ′ , … , m n ′ > 0 such that these compact open subgroups T i ⁢ ( m i ′ ) are all contained in 𝒢 and K ℓ and their images in 𝐓 ⁢ ( ℚ ) \ 𝐓 ⁢ ( 𝔸 f ) are all contained in 𝒢 . Then we get the identity as in the proposition. ∎

For simplicity, in the following we choose m 1 = m 2 = ⋯ = m n = m ≫ 0 and write

κ = 1 + ℓ m ⁢ ℤ ℓ .

We also put

K = K ℓ ⁢ K ℓ ⊂ 𝐇 ⁢ ( 𝔸 f ) .

For N ¯ = ( N 1 , … , N n ) ∈ ℕ n , write

u ⁢ ( ℓ - N ¯ ) = u 1 ⁢ ( ℓ - N 1 ) ⁢ ⋯ ⁢ u n ⁢ ( ℓ - N n ) .

For any map f : ( 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( 𝔸 f ) / K ) r → ℂ , define

A ⁢ ( f , u ⁢ ( t ¯ ) ) := ∫ 𝒢 f ⁢ ( Δ ⁢ ( h ) ⁢ g ⁢ Δ ⁢ ( u ⁢ ( t ¯ ) ) ) ⁢ 𝑑 μ 𝒢 ⁢ ( h ) for all ⁢ t ¯ = ( t 1 , … , t n ) ∈ ℚ ℓ n .

Corollary A.16.

Fix h ∈ T ⁢ ( A f ) . We have

A ⁢ ( f , u ⁢ ( t ¯ ) ) = ∫ 𝒢 f ⁢ ( Δ ⁢ ( h ) ⁢ g ⁢ Δ ⁢ ( u ⁢ ( ℓ - N ¯ ) ) ) ⁢ 𝑑 μ 𝒢 ⁢ ( h ) for all ⁢ t ¯ ∈ κ N ¯ .

In other words, the function A ⁢ ( f , u ⁢ ( t ¯ ) ) is constant on the variable t ¯ in the above specified domain κ N ¯ .

We can rewrite (A.4) as

lim N ¯ → + ∞ ⁡ A ⁢ ( f , u ⁢ ( ℓ - N ¯ ) ) = B ⁢ ( f ) := right-hand side of (A.4) .

We define the following objects:^[5]

SP := 𝐓 ⁢ ( ℚ ) \ 𝐇 ⁢ ( 𝔸 f ) ,

𝒳 := 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( 𝔸 f ) ,

𝒵 := 𝐇 ⁢ ( ℚ ) \ 𝐇 ⁢ ( 𝔸 f ) / 𝐇 1 ⁢ ( 𝔸 f ) .

Moreover, we have the following natural projection maps

SP → ℜ 𝒳 → 𝔄 𝒵 .

Note that 𝒳 and 𝒵 are both compact. Let 𝐇 ⁢ ( 𝔸 f ) act on the right on SP , 𝒳 , 𝒵 . Then these maps ℜ and 𝔄 are 𝐇 ⁢ ( 𝔸 f ) -equivariant. Similarly we define the following objects and maps:

SP K := SP / K , 𝒳 K := 𝒳 / K , 𝒵 K := 𝒵 / K , SP K → ℜ K 𝒳 K → 𝔄 K 𝒵 K .

Here 𝒳 K , 𝒵 K are finite sets. We define furthermore the following maps:

ℜ K r : SP K → 𝒳 K r , x ↦ ( ℜ K ⁢ ( g i ⁢ x ) ) i = 1 r ,

𝔄 K r : 𝒳 K r → 𝒵 K r , ( y i ) i = 1 r ↦ ( 𝔄 K ⁢ ( y i ) ) i = 1 r ,

𝔄 r : 𝒳 r → 𝒵 r , ( y i ) i = 1 r ↦ ( 𝔄 ⁢ ( y i ) ) i = 1 r .

Since 𝐇 1 ⁢ ( 𝔸 f ) acts transitively on each fiber of the map 𝔄 , for any z ∈ 𝒵 , there is a unique Borel measure μ z on 𝔄 - 1 ⁢ ( z ) invariant under 𝐇 1 ⁢ ( 𝔸 f ) .

Here is the main result of this appendix:

Theorem A.17.

For N 1 , … , N n ≫ 0 , one has

ℜ K r ⁢ ( 𝒢 ⁢ u ⁢ ( ℓ - N ¯ ) ) = ( 𝔄 K r ) - 1 ⁢ ( 𝔄 K r ∘ ℜ K r ⁢ ( 𝒢 ⁢ u ⁢ ( ℓ - N ¯ ) ) ) .

Proof.

We prove that right-hand side is contained in the left-hand side, the other direction being trivial. Let s ∈ ( 𝔄 K r ) - 1 ⁢ ( 𝔄 K r ⁢ ( ℜ K r ⁢ ( 𝒢 ⁢ u ⁢ ( ℓ - N ¯ ) ) ) ) and 𝕀 s : 𝒳 r → ℂ be the characteristic function of the pre-image of s under the projection map 𝒳 r → 𝒳 K r . We compute both sides of (A.4) for f = 𝕀 s . It is easy to see that A ⁢ ( 𝕀 s , u ⁢ ( ℓ - N ¯ ) ) is equal to the measure of the subset { h ∈ 𝒢 | ℜ K r ⁢ ( h ⁢ u ⁢ ( ℓ - N ¯ ) ) = s } of 𝒢 :

A ⁢ ( 𝕀 s , u ⁢ ( ℓ - N ¯ ) ) = μ 𝒢 ⁢ { h ∈ 𝒢 ∣ ℜ K r ⁢ ( h ⁢ u ⁢ ( ℓ - N ¯ ) ) = s } .

For any z = ( z 1 , … , z r ) ∈ 𝒵 r , write μ z for the product of the measures μ z i on 𝔄 - 1 ⁢ ( z i ) ( i = 1 , … , r ). Then we define

I ⁢ ( 𝕀 s , z ) := ∫ ( 𝔄 r ) - 1 ⁢ ( z ) 𝕀 s ⁢ 𝑑 μ z .

It is easy to see that I ⁢ ( 𝕀 s , ⋅ ) factors through the quotient 𝒵 r → 𝒵 K r . Write Ω ⁢ ( 𝒢 ) for the common cardinal of 𝒢 -orbits on 𝒵 K r . Then we have

B ⁢ ( 𝕀 s ) = I ⁢ ( 𝕀 s , 𝔄 K r ⁢ ( s ) ) Ω ⁢ ( 𝒢 ) .

Lemma A.18 shows that I ⁢ ( 𝕀 s , 𝔄 K r ⁢ ( s ) ) ≠ 0 . Moreover, 𝒳 K r is a finite set, thus there are only finitely many values for I ⁢ ( 𝕀 s , 𝔄 K r ⁢ ( s ) ) (none of which is zero). So for N 1 , … , N n ≫ 0 , we have

| A ⁢ ( 𝕀 s , u ⁢ ( ℓ - N ¯ ) ) - B ⁢ ( 𝕀 s ) | ≤ | B ⁢ ( 𝕀 s ) | 2 .

In particular, we have A ⁢ ( 𝕀 s , u ⁢ ( ℓ - N ¯ ) ) ≠ 0 , so for any s ∈ ( 𝔄 K r ) - 1 ⁢ ( 𝔄 K r ⁢ ( ℜ K r ⁢ ( 𝒢 ⁢ u ⁢ ( ℓ - N ¯ ) ) ) ) , there exists h ∈ 𝒢 such that ℜ K r ⁢ ( h ⁢ u ⁢ ( ℓ - N ¯ ) ) = s . ∎

Lemma A.18.

For s ∈ ( A K r ) - 1 ⁢ ( A K r ⁢ ( R K r ⁢ ( G ⁢ u ⁢ ( ℓ - N ¯ ) ) ) ) , we have

I ⁢ ( 𝕀 s , 𝔄 K r ⁢ ( s ) ) > 0 .

Proof.

We follow the proof in [4, Proposition 2.14].

We write z i = 𝔄 K ⁢ ( s i ) and set 𝕀 i : 𝒳 → { 0 , 1 } the characteristic function of the pre-image of s i under the projection map 𝒳 → 𝒳 K . Then we put

I ⁢ ( s i ) = ∫ 𝔄 - 1 ⁢ ( z i ) 𝕀 i ⁢ 𝑑 μ z i .

So one has

I ⁢ ( s ) := I ⁢ ( 𝕀 s , 𝔄 K r ⁢ ( s ) ) = ∏ i = 1 r I ⁢ ( s i ) .

Therefore to prove the lemma, it suffices to show that I ⁢ ( s i ) > 0 for any i = 1 , … , n .

Let z ∈ 𝒵 . The normalized Haar measure μ on 𝐇 1 ⁢ ( 𝔸 f ) induces the measure μ z on 𝔄 - 1 ⁢ ( z ) ⊂ 𝒳 . Note that μ z is the unique measure on 𝔄 - 1 ⁢ ( z ) such that for any compact open subgroup 𝐊 of 𝐇 1 ⁢ ( 𝔸 f ) and any x ∈ 𝔄 - 1 ⁢ ( z ) , one has

μ z ⁢ ( x ⁢ 𝐊 ) = μ ⁢ ( 𝐊 ) ♯ ⁢ 𝐊 x .

Note that 𝐊 x is compact and discrete, thus is a finite set. So μ z ⁢ ( x ⁢ 𝐊 ) > 0 . Moreover, for any g ∈ 𝐇 1 ⁢ ( 𝔸 f ) , the measure μ z ⁢ g ( ⋅ g ) = ( μ z ⁢ g ∘ g ) ( ⋅ ) on 𝔄 - 1 ⁢ ( z ) is equal to μ z . In other words, for any z 1 , z 2 ∈ 𝒵 ,

μ z 1 ⁢ ( 𝔄 - 1 ⁢ ( z 1 ) ) = μ z 2 ⁢ ( 𝔄 - 1 ⁢ ( z 2 ) ) .

Now for any x ∈ 𝒳 and z ∈ 𝒵 , we write

ϕ z ⁢ ( x ) = μ z ⁢ ( x ⁢ K ∩ 𝔄 - 1 ⁢ ( z ) ) .

Then the map 𝒵 → ℂ sending z to ϕ z ⁢ ( x ) factors through 𝒵 → 𝒵 K . Similarly, the map 𝒳 → ℂ sending x to ϕ z ⁢ ( x ) factors through 𝒳 → 𝒳 K . By definition we have

ϕ z ⁢ ( x ) ⁢ { = 0 , z ∉ 𝔄 ⁢ ( x ⁢ K ) , > 0 , z ∈ 𝔄 ⁢ ( x ⁢ K ) .

Thus for any i = 1 , … , r , we have

I ⁢ ( s i ) = ϕ z i ⁢ ( s i ) > 0 . ∎

A.4 Application to automorphic forms

Let 𝐇 , 𝐓 , 𝐇 j , 𝐔 j be as in Section A.1. Let K be a compact open subgroup of 𝐇 ⁢ ( 𝔸 f ) . We fix a ring A, an A-module M. Let g = ( g 1 , … , g r ) ∈ 𝐓 ⁢ ( 𝔸 f ) satisfy (A.2). Let 𝒢 be an open subgroup of 𝐓 ⁢ ( ℚ ) \ 𝐓 ⁢ ( 𝔸 f ) and write 𝒢 ~ its pre-image by the projection map 𝐓 ⁢ ( 𝔸 f ) → 𝐓 ⁢ ( ℚ ) \ 𝐓 ⁢ ( 𝔸 f ) . Then a similar argument as in [3, Corollary 5.2] gives:

Theorem A.19.

Let { β i } i = 1 r be a finite set of elements in A with β 1 ∈ A × . We assume that the composition map

𝒢 ~ → 𝐇 ⁢ ( 𝔸 f ) → 𝒵 K

is surjective. Consider a map f : H ⁢ ( Q ) \ H ⁢ ( A f ) / K → M which is not H 1 ⁢ ( A f ) -invariant (under right translation). Then, for any N 1 , … , N n ≫ 0 , there is an element h = h N ¯ ∈ G ~ such that

∑ i = 1 r β i ⁢ f ⁢ ( h ⁢ g i ⁢ u ⁢ ( ℓ - N ¯ ) ) ≠ 0 .

Proof.

By Theorem A.17, for any N 1 , … , N n ≫ 0 ,

ℜ K r ⁢ ( 𝒢 ⁢ u ⁢ ( ℓ - N ¯ ) ) = ( 𝔄 K r ) - 1 ⁢ ( 𝔄 K r ⁢ ( ℜ K r ⁢ ( 𝒢 ⁢ u ⁢ ( ℓ - N ¯ ) ) ) ) .

We fix one such N ¯ = ( N 1 , … , N n ) . By assumption, there are elements y 1 ≠ y 2 ∈ 𝒳 such that 𝔄 K ⁢ ( y 1 ) = 𝔄 K ⁢ ( y 2 ) and f ⁢ ( y 1 ) ≠ f ⁢ ( y 2 ) . Since 𝒢 ~ → 𝒵 K is surjective, we can choose x ∈ 𝒢 ~ such that

𝔄 K ⁢ ( y 1 ) = 𝔄 K ⁢ ( y 2 ) = 𝔄 K ⁢ ( ℜ K ⁢ ( x ⁢ u ⁢ ( ℓ - N ¯ ) ) ) .

Therefore for any finite subset { x i } i = 1 r of 𝔄 K - 1 ⁢ ( 𝔄 K ⁢ ( y 1 ) ) , we can find h 1 , h 2 ∈ 𝒢 ~ such that

ℜ K r ⁢ ( h i ⁢ x ⁢ u ⁢ ( ℓ - N ¯ ) ) = ( y i , x 2 , … , x r ) , i = 1 , 2 .

Thus one has

∑ i = 1 r β i ⁢ f ⁢ ( h 1 ⁢ g i ⁢ x ⁢ u ⁢ ( ℓ - N ¯ ) ) - ∑ i = 1 r β i ⁢ f ⁢ ( h 2 ⁢ g i ⁢ x ⁢ u ⁢ ( ℓ - N ¯ ) ) = β 1 ⁢ ( f ⁢ ( y 1 ) - f ⁢ ( y 2 ) ) ≠ 0 .

So we can take h = h N ¯ to be h 1 ⁢ x or h 2 ⁢ x . ∎

A.5 Examples

In this subsection we give some examples of ( 𝐇 , 𝐓 , 𝐇 j , 𝐔 j ) satisfying condition (A3). We give these explicit examples with the purpose in mind that they may be used directly in the theory of theta lifts.

A.5.1 Unitary groups

Let K be an imaginary quadratic number field with an embedding K ↪ ℚ ℓ . Let V = K n + 1 be a vector space over K of dimension n + 1 ≥ 2 equipped with a Hermitian form Q, which is represented, under the standard K-basis { E 1 , … , E n + 1 } of V, by a diagonal matrix Q = diag ⁡ ( δ 1 , … , δ n + 1 ) . Suppose that δ 1 , … , δ n + 1 > 0 . We take 𝐇 = U ⁢ ( V , Q ) the unitary group associated to ( V , Q ) . Then 𝐇 / K ≃ GL n + 1 / K as algebraic groups over K.

For any distinct basis elements E i , E j , we write U i , j for the unitary group associated to the Hermitian subspace ( K ⁢ ( E i , E j ) , Q ) of ( V , Q ) . Now we put

𝐇 j = U 1 , j + 1 , j = 1 , … , n .

We fix an isomorphism

ι : GL n + 1 ⁢ ( ℚ ℓ ) ≃ 𝐇 ⁢ ( ℚ ℓ ) ,

compatible with GL n + 1 ⁢ ( K ) ≃ 𝐇 ⁢ ( K ) under the embedding K ↪ ℚ ℓ such that 𝐇 j ⁢ ( ℚ ℓ ) is mapped isomorphically to the subgroup of GL n + 1 ⁢ ( ℚ ℓ ) consisting of matrices of the following form

( a b 1 j - 1 c d 1 n - j ) with ⁢ ( a b c d ) ∈ GL 2 ⁢ ( ℚ ℓ ) .

Fix isomorphisms

ι j : GL 2 ⁢ ( ℚ ℓ ) ≃ 𝐇 j ⁢ ( ℚ ℓ )

such that ( ι - 1 ∘ ι j ) ⁢ ( U 2 ⁢ ( ℚ ℓ ) ) consists of unipotent upper triangular matrices. We then take the unipotent subgroup 𝐔 j ⊂ 𝐇 j ⁢ ( ℚ ℓ ) to be the image of U 2 ⁢ ( ℚ ℓ ) by ι j . It is easy to see that these groups 𝐇 j 1 ⁢ ( ℚ ℓ ) generate 𝐇 1 ⁢ ( ℚ ℓ ) . Finally, we take

𝐓 = ∏ i = 1 n + 1 U ⁢ ( K ⁢ E i , Q ) .

One verifies that ( 𝐇 , 𝐓 , 𝐇 1 , … , 𝐇 n , 𝐔 1 , … , 𝐔 n ) satisfies condition (A3).

A.5.2 Even spin groups

Suppose that -1 has a square root in ℚ ℓ , which we denote by 𝐢 .

Let ( V , Q ) be a vector space over ℚ of even dimension 2 ⁢ n ≥ 4 such that under the standard basis { E 1 , E 2 , … , E 2 ⁢ n } of V, Q is represented by a diagonal matrix Q = diag ⁡ ( δ 1 , … , δ 2 ⁢ n ) . In the following, for 1 ≤ i 1 < i 2 < ⋯ < i k ≤ 2 ⁢ n , we write

δ i 1 , … , i k = δ i 1 ⁢ ⋯ ⁢ δ i k ,

E i 1 , … , i k = E i 1 ⊗ ⋯ ⊗ E i k .

We put the following conditions on δ 1 , … , δ 2 ⁢ n :

δ 1 , … , δ 2 ⁢ n > 0 ,
δ 1 , … , δ 2 ⁢ n ∈ ( ℚ ℓ × ) 2 ,
δ 1 , 2 , 3 , 4 , δ 1 , 2 , 5 , 6 , … , δ 1 , 2 , 2 ⁢ n - 1 , 2 ⁢ n ∈ ( ℚ × ) 2 .

We fix h i ∈ ℚ ℓ × ( i = 1 , … , 2 ⁢ n ) such that

h i 2 = δ i - 1 .

Then we take 𝐇 = GSpin ⁢ ( V , Q ) . More precisely, write C ⁢ ( V ) for the even degree part of the Clifford algebra

Cliff ⁢ ( V ) = ⊕ k = 0 ∞ V ⊗ k / 〈 v ⊗ v - Q ⁢ ( v ) | v ∈ V 〉

associated to ( V , Q ) . Then 𝐇 consists of those units v ∈ C ⁢ ( V ) × such that v ⁢ V = V ⁢ v . For any distinct basis elements E i 1 , E i 2 , … , E i k , we write C i 1 , … , i k for the even degree part of the Clifford algebra associated to the quadratic subspace ( ℚ ⁢ ( E i 1 , … , E i k ) , Q ) of ( V , Q ) . Then we have an isomorphism of ℚ ℓ -algebras:

M 2 ( ℚ ℓ ) ≃ C i , j , k ⊗ ℚ ℚ ℓ , ( 0 1 0 0 ) ↦ ( h i ⁢ E i + h j ⁢ 𝐢 ⁢ E j ) ⁢ ( h k ⁢ E k ) 2 = : e i , j , k + ,

( 0 0 1 0 ) ↦ - ( h i ⁢ E i - h j ⁢ 𝐢 ⁢ E j ) ⁢ ( h k ⁢ E k ) 2 = : e i , j , k - ,

( a 0 0 1 ) ↦ 1 + a - 1 4 ⁢ ( h i ⁢ E i + h j ⁢ 𝐢 ⁢ E j ) ⁢ ( h i - h j ⁢ 𝐢 ⁢ E j ) , a ∈ ℚ ℓ ,

( a 0 0 1 a ) ↦ 1 a ( 1 + a 2 - 1 4 ( h i E i + h j 𝐢 E j ) ( h i E i - h j 𝐢 E j ) ) = : τ i , j ( a ) , a ∈ ℚ ℓ × .

We have the following identities:

( 1 + v ⁢ e i 1 , i 2 , k + ) ⁢ ( 1 ± v ⁢ e i 1 , i 2 , l + ) = 1 + v ⁢ ( e i 1 , i 2 , k + ± e i 1 , i 2 , l + ) , i 1 , i 2 , k , l ∈ { 1 , … , 2 ⁢ n } ⁢ distinct ,

τ i 1 , i 2 ⁢ ( a ) ⁢ ( 1 + v ⁢ e i 1 , i 2 , k ± ) ⁢ τ i 1 , i 2 ⁢ ( a ) - 1 = 1 + a ± 2 ⁢ v ⁢ e i 1 , i 2 , k ± , i 1 , i 2 , k ∈ { 1 , … , 2 ⁢ n } ⁢ distinct .

For k = 2 , 3 , … , n , we fix a positive square root δ 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k of δ 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k in ℚ and we write

𝐞 k = 1 2 ⁢ ( 1 - E 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k δ 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k ) ,

which is a central idempotent element in C 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k . One verifies that both C 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k ⁢ 𝐞 k and C 1 , 2 , 2 ⁢ k - 1 , 2 ⁢ k ⁢ ( 1 - 𝐞 k ) are central simple algebras over ℚ of dimension 4. Then we put

𝐇 j = { ( C 1 , 2 , 3 , 4 ⁢ ( 1 - 𝐞 2 ) ) × + 𝐞 2 , j = 1 , ( C 1 , 2 , 2 ⁢ j - 1 , 2 ⁢ j ⁢ 𝐞 j ) × + ( 1 - 𝐞 j ) , j = 2 , … , n .

The unipotent subgroups 𝐔 j are given as follows:

𝐔 j = { ( 1 + ℚ ℓ ⁢ e 1 , 2 , 3 + ) ⁢ ( 1 - 𝐞 2 ) + 𝐞 2 = 1 + ℚ ℓ ⁢ ( e 1 , 2 , 3 + - 𝐢 ⁢ e 1 , 2 , 4 + ) , j = 1 , ( 1 + ℚ ℓ ⁢ e 1 , 2 , 2 ⁢ j - 1 + ) ⁢ 𝐞 j + ( 1 - 𝐞 j ) = 1 + ℚ ℓ ⁢ ( e 1 , 2 , 2 ⁢ j - 1 + + 𝐢 ⁢ e 1 , 2 , 2 ⁢ j + ) , j = 2 , … , n .

Moreover, it is easy to see these 𝐔 j commute with each other. The maximal torus is given by

𝐓 = ∏ k = 1 n C 2 ⁢ k - 1 , 2 ⁢ k × .

Thus the conjugate action of τ 2 ⁢ i - 1 , 2 ⁢ i ⁢ ( ℚ ℓ ) on 𝐔 k is given as follows:

Ad τ 2 ⁢ i - 1 , 2 ⁢ i ⁢ ( a ) ⁢ ( ( 1 + v ⁢ e 1 , 2 , 3 + ) ⁢ ( 1 - 𝐞 2 ) + 𝐞 2 ) = { ( 1 + a 2 ⁢ v ⁢ e 1 , 2 , 3 + ) ⁢ ( 1 - 𝐞 2 ) + 𝐞 2 , i ∈ { 1 , 2 } , ( 1 + v ⁢ e 1 , 2 , 3 + ) ⁢ ( 1 - 𝐞 2 ) + 𝐞 2 , i ∈ { 3 , … , n } ,

Ad τ 1 , 2 ⁢ ( a ) ⁢ ( ( 1 + v ⁢ e 1 , 2 , 2 ⁢ k - 1 + ) ⁢ 𝐞 k + ( 1 - 𝐞 k ) ) = ( 1 + a 2 ⁢ v ⁢ e 1 , 2 , 2 ⁢ k - 1 + ) ⁢ 𝐞 k + ( 1 - 𝐞 k ) , k ∈ { 2 , … , n } ,

Ad τ 2 ⁢ i - 1 , 2 ⁢ i ⁢ ( a ) ⁢ ( ( 1 + v ⁢ e 1 , 2 , 2 ⁢ k - 1 + ) ⁢ 𝐞 k + ( 1 - 𝐞 k ) ) = { ( 1 + a - 2 ⁢ v ⁢ e 1 , 2 , 2 ⁢ k - 1 + ) ⁢ 𝐞 k + ( 1 - 𝐞 k ) , i = k ∈ { 2 , … , n } , ( 1 + v ⁢ e 1 , 2 , 2 ⁢ k - 1 + ) ⁢ 𝐞 k + ( 1 - 𝐞 k ) , i ≠ k ∈ { 2 , … , n } .

One verifies easily that these 𝐇 1 1 ⁢ ( ℚ ℓ ) , … , 𝐇 n 1 ⁢ ( ℚ ℓ ) generate 𝐇 1 ⁢ ( ℚ ℓ ) : indeed, the above formula shows that the opposite root subgroups 𝐔 1 - , … , 𝐔 n - of 𝐔 1 , … , 𝐔 n (with respect to 𝐓 ⁢ ( ℚ ℓ ) ) are contained in the subgroup of 𝐇 1 ⁢ ( ℚ ℓ ) generated by 𝐇 1 1 ⁢ ( ℚ ℓ ) , … , 𝐇 n 1 ⁢ ( ℚ ℓ ) . As 𝐇 1 ⁢ ( ℚ ℓ ) is simply-connected, we deduce that 𝐇 1 1 ⁢ ( ℚ ℓ ) , … , 𝐇 n 1 ⁢ ( ℚ ℓ ) generate 𝐇 1 ⁢ ( ℚ ℓ ) . One checks easily that the remaining part of condition (A3) are also satisfied.

A.5.3 Odd spin groups

As in Section A.5.2, we suppose that -1 has a square root in ℚ ℓ , which we denote by 𝐢 .

Let ( V , Q ) be a vector space of odd dimension 2 ⁢ n + 1 ≥ 3 such that under the standard basis { E 0 , … , E 2 ⁢ n } of V, the quadratic form Q is represented by the diagonal matrix diag ⁡ ( δ 0 , δ 1 , … , δ 2 ⁢ n ) .

As in the preceding example, we put the following conditions on δ 0 , δ 1 , … , δ 2 ⁢ n :

δ 0 , … , δ 2 ⁢ n > 0 ,
δ 0 , … , δ 2 ⁢ n ∈ ( ℚ ℓ × ) 2 ,
δ 1 , 2 , 3 , 4 , δ 1 , 2 , 5 , 6 , … , δ 1 , 2 , 2 ⁢ n - 1 , 2 ⁢ n ∈ ( ℚ × ) 2 .

Using the notations from the preceding example, we take 𝐇 = GSpin ⁢ ( V , Q ) and

𝐇 j = { C 0 , 1 , 2 × , j = 1 , ( C 1 , 2 , 2 ⁢ j - 1 , 2 ⁢ j ⁢ 𝐞 i ) × + ( 1 - 𝐞 j ) , j = 2 , … , n .

The unipotent subgroups 𝐔 j are given as follows:

𝐔 j = { 1 + ℚ ℓ ⁢ e 1 , 2 , 0 + , j = 1 , ( 1 + ℚ ℓ ⁢ e 1 , 2 , 2 ⁢ j - 1 + ) ⁢ 𝐞 j , j = 2 , … , n .

One checks that these 𝐔 j commute with each other. The maximal torus is given by

𝐓 = ∏ k = 1 n C 2 ⁢ k - 1 , 2 ⁢ k × .

Thus the conjugate action of τ 2 ⁢ i - 1 , 2 ⁢ i ⁢ ( ℚ ℓ ) on 𝐔 1 is given as follows:

Ad τ 2 ⁢ i - 1 , 2 ⁢ i ⁢ ( a ) ⁢ ( 1 + v ⁢ e 1 , 2 , 0 + ) = { 1 + a 2 ⁢ v ⁢ e 1 , 2 , 0 + , i = 1 , 1 + v ⁢ e 1 , 2 , 0 + i ∈ { 2 , … , n } ,

and their actions on 𝐔 2 , … , 𝐔 n are the same as in the preceding example. The same reasoning as above shows that 𝐇 1 1 ⁢ ( ℚ ℓ ) , … , 𝐇 n 1 ⁢ ( ℚ ℓ ) generate 𝐇 1 ⁢ ( ℚ ℓ ) .

A.5.4 Symplectic groups

We write J 1 = ( 0 1 - 1 0 ) and for a positive integer n, we write

J 2 ⁢ n = diag ⁡ ( J 1 , … , J 1 ) ∈ GL 2 ⁢ n .

We define GSp 2 ⁢ n to be the group scheme over ℤ consisting of matrices X ∈ GL 2 ⁢ n such that X ⁢ J 2 ⁢ n ⁢ X t = μ ⁢ ( X ) ⁢ J 2 ⁢ n with μ ⁢ ( X ) ∈ 𝔾 m .

Fix an integer n > 1 and 𝐁 a definite quaternion algebra over ℚ , which is split at the place ℓ of ℚ as in Example A.2 and n > 1 an integer. Let ∗ : 𝐁 → 𝐁 be a main involution and for an n × n -matrix g = ( g i , j ) i , j = 1 n with entries in 𝐁 , write g ∗ to be ( g j , i ∗ ) i , j = 1 n . Then we define a quaternionic unitary group

𝐇 = { g ∈ GL n ⁢ ( B ) | g ⁢ g ∗ = μ ⁢ ( g ) ⋅ 1 n ⁢ for some ⁢ μ ⁢ ( g ) ∈ ℚ × } .

We view ℍ as an algebraic group over ℚ . Since 𝐁 is split at ℓ , we have an isomorphism of ℚ ℓ -algebras

ι : M 2 ⁢ ( ℚ ℓ ) ≃ 𝐁 ⊗ ℚ ℓ ,

which induces an isomorphism

ι : GSp 2 ⁢ n ⁢ ( ℚ ℓ ) ≃ 𝐇 ⁢ ( ℚ ℓ ) .

We fix then a maximal commutative subalgebra K of 𝐁 , which is a quadratic field extension of ℚ . We fix a ℚ -basis of 𝐁

𝐁 = ℚ ⁢ ( 1 , 𝐢 , 𝐣 , 𝐤 ) .

We assume that 𝐤 = 𝐢𝐣 = - 𝐣𝐢 , 𝐢 2 , 𝐣 2 , 𝐤 2 ∈ ℚ × and K = ℚ ⁢ [ 𝐢 ] splits at ℓ .^[6] We fix embeddings ι j : 𝐁 × ↪ 𝐇 as follows ( j = 1 , … , n ):

ι j : 𝐁 × → 𝐇 , x = a + b ⁢ 𝐢 + c ⁢ 𝐣 + d ⁢ 𝐤 ↦ { ( x 1 n - 1 ) , j = 1 , ( a + b ⁢ 𝐢 c ⁢ 𝐣 + d ⁢ 𝐤 1 j - 2 - c ⁢ 𝐣 - d ⁢ 𝐤 a + b ⁢ 𝐢 1 n - j ) , j = 2 , … , n .

Then set 𝐇 j = ι j ⁢ ( 𝐁 × ) . We fix an isomorphism of ℚ ℓ -algebras

𝐁 ⊗ ℚ ℓ → M 2 ⁢ ( ℚ ℓ ) , 𝐢 ⊗ 1 ↦ ( 𝐢 0 0 - 𝐢 ) , 𝐣 ⊗ 1 ↦ ( 0 𝐣 2 1 0 ) , 𝐤 ⊗ 1 ↦ ( 0 𝐢𝐣 2 - 𝐢 0 ) .

The unipotent subgroups 𝐔 j are given by

𝐔 j = ( ι ∘ ι j ∘ ι - 1 ) ⁢ ( U 2 ) .

Clearly, these unipotent subgroups commute with each other. The maximal torus 𝐓 of 𝐇 is given by

𝐓 = { g = diag ⁡ ( g 1 , … , g n ) ∈ 𝐇 | g 1 , … , g n ∈ K × ⁢ with ⁢ g 1 ⁢ g 1 ∗ = g 2 ⁢ g 2 ∗ = ⋯ = g n ⁢ g n ∗ = μ ⁢ ( g ) ∈ ℚ × } .

So in the induced isomorphism 𝐇 ⁢ ( ℚ ℓ ) ≃ GSp 2 ⁢ n ⁢ ( ℚ ℓ ) , 𝐓 ⁢ ( ℚ ℓ ) is mapped to the subgroup of diagonal matrices. For any a ∈ ℚ ℓ × and any j = 1 , … , n , we write

τ j ⁢ ( a ) = diag ⁡ ( 1 2 ⁢ ( j - 1 ) , a , 1 / a , 1 2 ⁢ ( n - j ) ) ∈ GSp 2 ⁢ n ⁢ ( ℚ ℓ ) .

For any j = 1 , … , n , we write

u j ⁢ ( t ) = ( ι ∘ ι j ∘ ι - 1 ) ⁢ ( ( 1 t 0 1 ) ) ∈ 𝐇 ⁢ ( ℚ ℓ ) .

Then one checks easily the conjugate action Ad of τ j ⁢ ( a ) on 𝐔 i is given by

Ad τ j ⁢ ( a ) ⁢ ( u i ⁢ ( v ) ) = { u i ⁢ ( a 2 ⁢ v ) , i = j = 1 , u i ⁢ ( a ⁢ v ) , i ≠ j = 1 , u i ⁢ ( a - 1 ⁢ v ) , i = j ≠ 1 , u i ⁢ ( v ) , i ≠ j ≠ 1 .

Using the above formulas and the argument similar to the preceding examples, one verifies easily that 𝐇 1 1 ⁢ ( ℚ ℓ ) , … , 𝐇 n 1 ⁢ ( ℚ ℓ ) generate 𝐇 1 ⁢ ( ℚ ℓ ) and condition (A3) is satisfied.

Acknowledgements

We would like to thank Massimo Bertolini, Christophe Cornut, Haruzo Hida, Ming-Lun Hsieh and Jacques Tilouine for a lot of useful and stimulating discussions concerning earlier drafts of this article. We thank Zheng Liu in particular for careful reading which uncovers an error in an early version of the article. At last we would like to express our thanks to the referee for his/her many useful comments, suggestions and questions, which help improve a lot the structure and the readability of this article.

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Received: 2024-11-05

Revised: 2025-07-11

Published Online: 2025-08-01

Published in Print: 2026-05-01

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Schlagwörter für diesen Artikel

theta correspondence; ergodic theory

Non-vanishing mod p of theta lifts for (O2n+1, Mp4n )

Artikel

Abstract

A Toric orbits of unipotent elements

A.1 Basic set-up

(A3).

Remark A.1.

Examples A.2.

A.2 Ratner’s theorems and commensurability

A.2.1 SL(2) case

Theorem A.3.

Remark A.4.

Theorem A.5.

Theorem A.6.

Remark A.7.

A.2.2 Adelic reformulation

Lemma A.8.

Proof.

Corollary A.9.

A.2.3 Single-copy case

Lemma A.10.

Proof.

Remark A.11.

Theorem A.12.

Proof.

A.2.4 Multi-copy case

Lemma A.13.

Proof.

Theorem A.14.

A.3 Main result

Proposition A.15.

Proof.

Corollary A.16.

Theorem A.17.

Proof.

Lemma A.18.

Proof.

A.4 Application to automorphic forms

Theorem A.19.

Proof.

A.5 Examples

A.5.1 Unitary groups

A.5.2 Even spin groups

A.5.3 Odd spin groups

A.5.4 Symplectic groups

Acknowledgements

References

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Non-vanishing mod p of theta lifts for (O_2n+1, Mp_4n)