The Tate module of a simple abelian variety of type IV

New York Journal of Mathematics

New York J. Math.27(2021) 1240–1257.

The Tate module of a simple abelian variety of type IV

Grzegorz Banaszak and Aleksandra Kaim-Garnek

Abstract. The aim of this paper is to investigate the Galois module struc- ture of the Tate module of an abelian variety defined over a number field. We focus on simple abelian varieties of type IV in Albert classification. We de- scribe explicitly the decomposition of the𝒪_𝜆[𝐺_𝐹]-module𝑇_𝜆(𝐴)into compo- nents that are rationally and residually irreducible. Moreover these compo- nents are non-degenerate, hermitian modules that rationally and residually are non-degenerate, hermitian vector spaces.

Contents

1. Introduction 1240

2. Ring of endomorphisms of an abelian variety of type IV 1243 3. Weil pairing of an abelian variety of type IV 1246

4. Main theorem for𝑑 ≤ 2 1252

5. Main theorem for𝑑 > 2 1254

References 1256

1. Introduction

Let𝐴be an abelian variety of dimension𝑔over a number field𝐹.Letℛ ∶=

End_𝐹(𝐴).Put𝐷 ∶= ℛ ⊗_ℤℚand let𝐸be the center of𝐷.The ringℛis an order in𝐷.Becauseℛis finitely generatedℤ-module thenℛ ∩ 𝐸 = 𝒪⁰_𝐸is an order in 𝒪_𝐸.Throughout the paper we fix a polarization of𝐴.Let𝑙be a prime number and let𝑇_𝑙(𝐴)be the Tate module of𝐴. Let𝐺_𝐹 ∶= Gal(𝐹∕𝐹)and let

𝜌_𝑙 ∶ 𝐺_𝐹 → 𝐺𝐿(𝑇_𝑙(𝐴)) be the𝑙-adic representation associated with𝐴.

From now on, we assume thatℛ is defined over𝐹,i.e. ℛ = End_𝐹(𝐴)so 𝐷 = End_𝐹(𝐴) ⊗_ℤℚ.

Received November 15, 2020.

2020Mathematics Subject Classification. 11F80, 11Gxx, 16K20.

Key words and phrases. Abelian variety, Tate module, Galois𝑙-adic representation.

Acknowledgement: The authors would like to thank the referee for valuable comments, sug- gestions and corrections which improved the exposition of the paper.

ISSN 1076-9803/2021

1240

In this paper we also assume that𝐴is simple, hence𝐷is a division algebra of finite dimension overℚwith a positive involution^′[14, p. 193-203]. Let𝐸₀be the subfield of elements of𝐸 fixed by^′. We put𝑑² ∶= [𝐷 ∶ 𝐸], 𝑒 ∶= [𝐸 ∶ ℚ]

and𝑒₀∶= [𝐸₀∶ ℚ].

Recall that, due to A. A. Albert, simple abelian varieties can be classified according to the type of their endomorphism algebra (see: [1] and [14, Theorem 2, p. 201-203]):

TYPE I: 𝐷 = 𝐸 = 𝐸₀is a totally real field.

TYPE II: 𝐸 = 𝐸₀is a totally real field and𝐷 is a quaternion division algebra overℚsuch that𝐷 ⊗_𝐸

0 ℝ ≅ 𝑀₂(ℝ)for any embedding𝜎 ∶ 𝐸₀ → ℝ. TYPE III: 𝐸 = 𝐸₀is a totally real field and𝐷is a quaternion division algebra

overℚsuch that𝐷 ⊗_𝐸0 ℝ ≅ ℍfor any embedding𝜎 ∶ 𝐸₀→ ℝ.

TYPE IV: 𝐸₀is a totally real field,𝐸is a totally imaginary quadratic extension of𝐸₀and𝐷is a division algebra overℚsuch that𝐷 ⊗_𝐸0ℝ ≅ 𝑀_𝑑(ℂ)for any embedding𝜎 ∶ 𝐸₀→ ℝ.

If𝐴is of type I then𝑑 = 1. If𝐴is of type II or III then𝑑 = 2, and if𝐴is of type IV then𝑑 ≥ 1can be arbitrary. Moreover, if𝐴is a simple abelian variety of type IV then𝐸is a quadratic imaginary extension of a totally real field𝐸₀so 𝑒 = 2𝑒₀cf. [14, Theorem 2, p. 201-203].

Let𝜆be a prime ideal in𝒪_𝐸 dividing𝑙. Let𝒪_𝜆be the completion of𝒪_𝐸at𝜆, 𝐸_𝜆 ∶= Frac(𝒪_𝜆)and𝑘_𝜆 ∶= 𝒪_𝜆∕𝜆. Observe that for each𝑙 ∶

𝐸_𝑙 ∶= 𝐸 ⊗_ℚℚ_𝑙 =∏

𝜆|𝑙

𝐸_𝜆 and 𝒪_𝐸𝑙 ∶= 𝒪_𝐸⊗_ℤℤ_𝑙 =∏

𝜆|𝑙

𝒪_𝜆. (1.1) For𝑙 ∤ [𝒪_𝐸 ∶ 𝒪⁰_𝐸], we have𝒪⁰_𝐸 ⊗_ℤℤ_𝑙 = 𝒪_𝐸⊗_ℤℤ_𝑙.Hence, for such an𝑙, the ring𝒪_𝐸𝑙 acts on𝑇_𝑙(𝐴)and we put𝑇_𝜆(𝐴) ∶= 𝑇_𝑙(𝐴) ⊗_𝒪

𝐸𝑙 𝒪_𝜆.Hence, for each 𝑙 ∤ [𝒪_𝐸 ∶ 𝒪⁰_𝐸] ∶

𝑇_𝑙(𝐴) =⨁

𝜆|𝑙

𝑇_𝜆(𝐴). (1.2)

The aim of this paper is to describe explicitly the decomposition of the𝒪_𝜆[𝐺_𝐹]- module𝑇_𝜆(𝐴)for a simple abelian variety𝐴of type IV into components that are rationally and residually irreducible. We also show that these components are compatible with corresponding non-degenerate, hermitian forms. This work is a continuation of the research in [2], [3] and [4] on the Galois𝑙-adic repre- sentations for abelian varieties of types I, II and III. In the papers loc. cit., the first author with W. Gajda and P. Krasoń showed that𝑇_𝜆(𝐴)has the following decomposition for𝑙 ≫ 0:

𝑇_𝜆(𝐴) ≅ 𝒲_𝜆(𝐴)^𝑑,

where𝒲_𝜆(𝐴)is a free𝒪_𝜆-module of rank^2𝑔_𝑒𝑑,with non-degenerate bilinear,𝐺_𝐹- equivariant form

𝜓_𝜆 ∶ 𝒲_𝜆(𝐴) × 𝒲_𝜆(𝐴) → 𝒪_𝜆

such that𝑊_𝜆(𝐴) ∶= 𝒲_𝜆(𝐴) ⊗_𝒪𝜆 𝐸_𝜆 is an absolutely irreducible𝐺_𝐹-module with a non-degenerate,𝐺_𝐹-equivariant bilinear form𝜓_𝜆⁰ ∶= 𝜓_𝜆 ⊗_𝒪𝜆 𝐸_𝜆 (resp.

𝒲_𝜆(𝐴) = 𝒲_𝜆(𝐴) ⊗_𝒪𝜆𝑘_𝜆 is an absolutely irreducible𝐺_𝐹-module with a non- degenerate,𝐺_𝐹-equivariant bilinear form𝜓_𝜆 ∶= 𝜓_𝜆⊗_𝒪𝜆𝑘_𝜆). For type I and II, the forms𝜓_𝜆,𝜓_𝜆⁰and𝜓_𝜆are alternating and for type III the forms𝜓_𝜆,𝜓⁰_𝜆and𝜓_𝜆 are symmetric.

For the case of abelian varieties of type II, this result extends integrally and residually the main result of [8, Theorem A] by W. C. Chi.

The Galois module structure of𝑇_𝑙(𝐴)for abelian varieties𝐴of types I, II and III has been widely investigated as well as Galois module structure of 𝑇_𝑙(𝐴) for abelian varieties𝐴of type IV with𝐷commutative (in particular see [9] for type IV). Such results are useful for current research. Results in [2], [3] and [4]

also found a variety of applications eg. [5], [6], [7], [10], [16], [17], [18] just to mention a few recent papers. Similarly as in [2], [3] and [4], we expect to prove the Mumford-Tate conjecture for some families of abelian varieties of type IV based on results of this paper.

In this paper we address all abelian varieties of type IV, especially those with 𝐷noncommutative. In general, endomorphism algebras of abelian varieties of type IV are much more complicated than endomorphism algebras of abelian varieties of types I, II and III. Indeed, the degree of𝐷 over𝐸may be arbitrary and the standard involution acts nontrivially on its center which is CM field, c.f.

[14, Theorem 2, p. 201-203]. Nevertheless, we obtain new results for abelian varieties of type IV (see Theorem1.1below) showing striking similarity with corresponding results for abelian varieties of types I, II and III discussed above.

In Section 2, we describe as explicitly as possible (Lemma2.1) the endomor- phism algebra𝐷of a simple abelian variety of type IV and its splitting field𝐿to obtain an𝐿-algebra isomorphism:

𝑠 ∶ 𝐷 ⊗_𝐸0𝐿⁺= 𝐷 ⊗_𝐸𝐿,→ 𝑀^∼ _𝑑(𝐿).

Lemma2.1is an arithmetic refinement of computations in D. Mumford’s book [14,§Application I, Step IV]. It is one of the main technical devices in our paper.

Observe how Lemma2.1 significantly differs from corresponding lemma ([4, Lemma 2.11]) for abelian varieties of type III. Based on this, we also obtain an S-integral splitting of the algebraℛ_𝑆 ∶= ℛ ⊗_ℤℤ_𝑆 at the end of this section.

This result and construction of the finite set𝑆can be found in Corollary2.3.

We also construct a positive involution𝑥 ↦ 𝑥^∗ of𝐷 which has the extension 𝑋 ↦ 𝑋^∗ ∶= 𝑋^𝑇𝑟to the ring𝑀_𝑑(𝐿)via splitting𝑠(see Lemma2.1for details).

In Section 3, we carefully investigate the Tate module of abelian variety of type IV based on results of Section 2. Namely, adding a few assumptions on𝑆and working as far as possible𝑆-integrally, we eventually construct non-degenerate,

𝐺_𝐹-equivariant hermitian forms (see Lemma3.5) to obtain, in Sections 4 and 5, our main result as follows.

Theorem 1.1(Theorems4.3,5.2). Let𝐴be an abelian variety of type𝐼𝑉.Let𝑙 be a prime outside of a finite set𝑆. Let𝜆|𝑙be a prime of𝒪_𝐸 such that𝜆is inert over𝜆₀∶= 𝜆 ∩ 𝒪_𝐸0and𝜆splits completely in𝒪_𝐿. The𝒪_𝜆[𝐺_𝐹]-module𝑇_𝜆(𝐴)has the following decomposition:

𝑇_𝜆(𝐴) ≅ 𝒲_𝜆(𝐴)^𝑑, where𝒲_𝜆(𝐴)is a free𝒪_𝜆-module of rank ^2𝑔

𝑒𝑑 with a non-degenerate, hermitian, 𝐺_𝐹-equivariant form

𝜓_𝜆 ∶ 𝒲_𝜆(𝐴) × 𝒲_𝜆(𝐴) → 𝒪_𝜆

such that 𝑊_𝜆(𝐴) ∶= 𝒲_𝜆(𝐴) ⊗_𝒪𝜆 𝐸_𝜆 is an absolutely irreducible 𝐺_𝐹-module with a non-degenerate, hermitian,𝐺_𝐹-equivariant form𝜓⁰_𝜆 ∶= 𝜓_𝜆⊗_𝒪𝜆𝐸_𝜆(resp.

𝒲_𝜆(𝐴) ∶= 𝒲_𝜆(𝐴) ⊗_𝒪𝜆 𝑘_𝜆 is an absolutely irreducible𝐺_𝐹-module with a non- degenerate, hermitian,𝐺_𝐹-equivariant form𝜓_𝜆 ∶= 𝜓_𝜆⊗_𝒪𝜆𝑘_𝜆).

Remark 1.2. The restrictions on prime𝑙in Theorem1.1 result mainly from complexity of the endomorphism algebra of𝐴.

2. Ring of endomorphisms of an abelian variety of type IV

Let𝐴be a simple abelian variety of type IV satisfying all assumptions stated in Section 1. Let ^′ be the standard Rosati involution on𝐷.This is a positive involution. Any other positive involution∗of 𝐷 is of the form𝑥^∗ = 𝛾𝑥^′𝛾⁻¹ with𝛾 ∈ 𝐷and𝛾^′= 𝛾.There exists a positive involution𝑥^∗ = 𝛾𝑥^′𝛾⁻¹of𝐷and an isomorphism [14, Theorem 2, p. 201-203]

𝐷 ⊗_ℚℝ,→ 𝑀^∼ _𝑑(ℂ) × ⋯ × 𝑀_𝑑(ℂ)

⏟⎴⎴⎴⎴⎴⎴⏟⎴⎴⎴⎴⎴⎴⏟

𝑒₀copies

,

which carries this involution into(𝑋₁, … , 𝑋_𝑒0) ↦ (𝑋^𝑇𝑟₁ , … , 𝑋^𝑇𝑟_𝑒0).Under the above isomorphism𝛾 ⊗ 1maps to(𝐶₁, … , 𝐶_𝑒0), where each𝐶_𝑖 is a hermitian positive definite matrix. It follows that there exists a positive involution𝑥^∗ = 𝛾𝑥^′𝛾⁻¹of 𝐷and an isomorphism

𝐷 ⊗_𝐸0 ℝ,→ 𝑀^∼ _𝑑(ℂ), (2.1)

which carries the involution∗of𝐷into the involution𝑋 ↦ 𝑋^∗ ∶= 𝑋^𝑇𝑟by the isomorphism (2.1) and𝛾 ⊗ 1 ↦ 𝐶, where𝐶 is a hermitian positive definite matrix.

For our applications, we need an arithmetic refinement of the above state- ments as follows.

Lemma 2.1. There exist a finite Galois extension𝐿∕𝐸₀containing𝐸,an element 𝛾 ∈ 𝐷with𝛾^′= 𝛾and an𝐿-algebra isomorphism

𝑠 ∶ 𝐷 ⊗_𝐸

0𝐿⁺= 𝐷 ⊗_𝐸𝐿,→ 𝑀^∼ _𝑑(𝐿) (2.2) such that via this isomorphism the positive involution𝑥 ↦ 𝑥^∗ ∶= 𝛾𝑥^′𝛾⁻¹of𝐷 has the extension𝑋 ↦ 𝑋^∗ ∶= 𝑋^𝑇𝑟to the ring𝑀_𝑑(𝐿)and𝑠(𝛾 ⊗ 1) = 𝐶, where𝐶is a hermitian positive definite matrix. Here,𝐿⁺denotes the maximal real subfield of𝐿.Changing base toℝover𝐿⁺the isomorphism (2.2) naturally extends to an isomorphism of the form (2.1) with the same properties.

Proof. From [13, Theorem 16, Chap. 29], we know that𝐷 has maximal sub- fields which are splitting fields of degree𝑑over𝐸.Hence, let𝐿₀be a maximal subfield of degree𝑑 = [𝐿₀∶ 𝐸]such that𝐷 ⊗_𝐸 𝐿₀,→ 𝑀^∼ _𝑑(𝐿₀).Let𝐿₁∕𝐸₀be the Galois closure of𝐿₀∕𝐸₀.Naturally𝐷 ⊗_𝐸𝐿₁,→ 𝑀^∼ _𝑑(𝐿₁). Moreover, we obtain:

𝐷 ⊗_𝐸0𝐿₁⁺ = 𝐷 ⊗_𝐸 𝐸 ⊗_𝐸0𝐿₁⁺ = 𝐷 ⊗_𝐸 𝐿₁ ,→ 𝑀^∼ _𝑑(𝐿₁). (2.3) Now we argue similarly to [14, p. 199-200]. By the Skolem-Noether theorem, the Rosati involution on𝐷 ⊗_𝐸0𝐿₁⁺(acting trivially on𝐿⁺₁) extends to an involu- tion of𝑀_𝑑(𝐿₁)of the following form:

𝑋 ↦ 𝐴₁𝑋^∗𝐴₁⁻¹ (2.4)

with𝐴₁ ∈ 𝐺𝐿_𝑑(𝐿₁). Because (2.4) is an involution, we get𝐴^∗₁ = 𝜂 𝐴₁for an element𝜂 ∈ 𝐿^×₁ such that|𝜂| = 1.Let𝐿₂∕𝐸₀be the Galois closure of𝐿₁(𝜂

1 2)∕𝐸₀. We obtain:

𝐷 ⊗_𝐸0𝐿₂⁺ = 𝐷 ⊗_𝐸 𝐸 ⊗_𝐸0𝐿₂⁺ = 𝐷 ⊗_𝐸 𝐿₂ ,→ 𝑀^∼ _𝑑(𝐿₂). (2.5) If𝜂 ≠ 1observe that:

(𝜂

1

2𝐴₁)^∗ = 𝜂⁻

1

2𝐴^∗₁ = 𝜂⁻

1

2𝜂𝐴₁= 𝜂

1 2 𝐴₁, 𝜂

1

2𝐴₁𝑋^∗(𝜂

1

2𝐴₁)⁻¹= 𝐴₁𝑋^∗𝐴⁻¹₁ . Hence,𝐴₂∶= 𝜂

1

2𝐴₁ ∈ 𝑀_𝑑(𝐿₂)is a hermitian matrix and the Rosati involution on𝐷 ⊗_𝐸0 𝐿⁺₂ (acting trivially on𝐿₂⁺) extends to an involution of𝑀_𝑑(𝐿₂)of the following form:

𝑋 ↦ 𝐴₂𝑋^∗𝐴₂⁻¹. (2.6) Observe that 𝐴₂ is a fixed point of the involution (2.6). The set of elements in𝐷 ⊗_𝐸0 𝐿₂⁺fixed by this involution via (2.5) (equivalently fixed by the Rosati involution) is of the form𝑉 ⊗_𝐸0 𝐿⁺₂,where𝑉 is the𝐸₀-vector space𝑉 = {𝛼 ∈ 𝐷 ∶ 𝛼^′ = 𝛼}.Indeed, by primitive element theorem there is𝛿 ∈ 𝐿⁺₂ such that 𝐿⁺₂ = 𝐸₀(𝛿).Let𝑟 ∶= [𝐿⁺₂ ∶ 𝐸₀].Then every element of𝐷 ⊗_𝐸0𝐿⁺₂ is of the form

∑𝑟−1

𝑖=0 𝛼_𝑖⊗ 𝛿^𝑖 for some𝛼_𝑖 ∈ 𝐷.The element∑𝑟−1

𝑖=0 𝛼_𝑖⊗ 𝛿^𝑖is fixed by the Rosati involution if and only if

𝑟−1∑

𝑖=0

(𝛼^′_𝑖 − 𝛼_𝑖) ⊗ 𝛿^𝑖 = 0 and this occurs if and only if𝛼_𝑖^′= 𝛼_𝑖for each𝑖.

Let∑𝑟−1

𝑖=0 𝛼_𝑖⊗𝛿^𝑖 ∈ 𝑉⊗_𝐸0𝐿⁺₂ be the element sent via (2.5) to𝐴₂.Note that𝐸₀is dense inℝwith respect to the absolute value. Therefore, we can find elements 𝑒_𝑖 ∈ 𝐸₀,close enough to𝛿^𝑖 ∈ 𝐿₂⁺,such that the element

𝛼 ⊗ 1 =

𝑟−1∑

𝑖=0

𝛼_𝑖𝑒_𝑖⊗ 1 =

𝑟−1∑

𝑖=0

𝛼_𝑖 ⊗ 𝑒_𝑖

maps via (2.5) to𝐵₂such that𝐴₂𝐵^∗₂𝐴₂⁻¹= 𝐵₂and𝐴₃ ∶= 𝐵⁻¹₂ 𝐴₂is very close to unit matrixI_𝑑.Observe that𝐴₃is a hermitian matrix. Indeed, we have(𝐵⁻¹₂ )^∗= (𝐵^∗₂)⁻¹= 𝐴⁻¹₂ 𝐵⁻¹₂ 𝐴₂.Hence:

𝐴^∗₃ = (𝐵₂⁻¹𝐴₂)^∗ = 𝐴₂(𝐵⁻¹₂ )^∗ = 𝐴₂𝐴₂⁻¹𝐵⁻¹₂ 𝐴₂ = 𝐵⁻¹₂ 𝐴₂= 𝐴₃.

The hermitian matrix𝐴₃,being very close toI_𝑑,is positive definite. There exist a finite Galois extension𝐿₃∕𝐸₀with𝐿₂ ⊂ 𝐿₃,a unitary matrix𝑈 ∈ 𝐺𝐿_𝑑(𝐿₃) and a diagonal matrix𝐷₃∈ 𝐺𝐿_𝑑(𝐿⁺₃)with positive entries on the diagonal such that𝐴₃ = 𝑈 𝐷₃²𝑈^∗.Put𝐵₃ ∶= 𝑈 𝐷₃𝑈^∗ ∈ 𝐺𝐿_𝑑(𝐿₃).Observe that𝐵^∗₃ = 𝐵₃and 𝐴₃= 𝐵₃².By (2.5) we obtain:

𝐷 ⊗_𝐸0𝐿₃⁺ = 𝐷 ⊗_𝐸 𝐸 ⊗_𝐸0𝐿₃⁺ = 𝐷 ⊗_𝐸 𝐿₃ ,→ 𝑀^∼ _𝑑(𝐿₃). (2.7) Observe that the map:

𝑥 ↦ 𝑥^∗∶= 𝛼⁻¹𝑥^′𝛼 (2.8) is an involution of𝐷 ⊗_𝐸0𝐿⁺₃ and it extends via (2.7) to the following involution of𝑀_𝑑(𝐿₃) ∶

𝑋 ↦ 𝐴₃𝑋^∗𝐴₃⁻¹. (2.9) Now we put𝐿 ∶= 𝐿₃and𝛾 ∶= 𝛼⁻¹.Composing the isomorphism (2.7)𝑥 ↦ 𝑋 with the conjugation by𝐵₃,namely𝑋 ↦ 𝐵⁻¹₃ 𝑋𝐵₃, we obtain the isomorphism:

𝑠 ∶ 𝐷 ⊗_𝐸0 𝐿⁺ ,→ 𝑀^∼ _𝑑(𝐿), (2.10) 𝑠(𝑥) ∶= 𝐵₃⁻¹𝑋𝐵₃. (2.11) Observe that:

𝑠(𝑥^∗) = 𝑠(𝑥)^∗. (2.12) Indeed:

𝑠(𝑥^∗) = 𝑠(𝛾𝑥^′𝛾⁻¹) = 𝐵⁻¹₃ 𝐴₃𝑋^∗𝐴⁻¹₃ 𝐵₃= 𝐵₃𝑋^∗𝐵₃⁻¹= (𝐵⁻¹₃ 𝑋𝐵₃)^∗ = 𝑠(𝑥)^∗.

Hence the involution𝑥 ↦ 𝑥^∗ = 𝛾𝑥^′𝛾⁻¹ extends via (2.10) to the involution 𝑋 ↦ 𝑋^∗of𝑀_𝑑(𝐿).The last statement of the lemma follows because:

𝐷 ⊗_𝐸0ℝ = 𝐷 ⊗_𝐸𝐸 ⊗_𝐸0𝐿⁺⊗_𝐿⁺ℝ = 𝐷 ⊗_𝐸𝐿 ⊗_𝐿⁺ℝ,→ 𝑀^∼ _𝑑(𝐿) ⊗_𝐿⁺ℝ = 𝑀_𝑑(ℂ).

(2.13) Naturally, the involution𝑥 ↦ 𝑥^∗ = 𝛾𝑥^′𝛾⁻¹of𝐷extends to the involution𝑋 ↦ 𝑋^∗of𝑀_𝑑(ℂ).

The following diagram illustrates relations between consecutive extensions of fields𝐸₀and𝐸used in this proof.

𝐸 𝐿₀ 𝐿₁ 𝐿₂ 𝐿 ∶= 𝐿₃ ℂ

ℚ 𝐸₀ 𝐿⁺₀ 𝐿⁺₁ 𝐿⁺₂ 𝐿⁺₃ ℝ

𝑑

2 2 2 2 2

𝑒0

𝑒 2

Remark 2.2. Lemma2.1is useful for the proof of Proposition3.7which is cru- cial in the proof of Lemma4.1ultimately leading to the proof of Theorem1.1.

Recall that the ringℛ is a finitely generated freeℤ-module. Let𝒪⁰_𝐸

0 ∶=

ℛ ∩ 𝒪_𝐸0 and𝒪⁰_𝐸 ∶= ℛ ∩ 𝒪_𝐸.Then𝒪⁰_𝐸

0 is an order in𝒪_𝐸0and𝒪⁰_𝐸 is an order in 𝒪_𝐸.

Let𝑆be a set of primes ofℤcontaining prime numbers that divide the in- dexes[𝒪_𝐸

0 ∶ 𝒪⁰_𝐸

0]and[𝒪_𝐸 ∶ 𝒪⁰_𝐸].

Corollary 2.3. One can enlarge𝑆so that the primes not in𝑆are unramified in 𝒪_𝐿,all primes dividing the polarization degree of𝐴are in𝑆,the Rosati involution acts onℛ_𝑆 ∶= ℛ ⊗_ℤℤ_𝑆, 𝛾 ∈ ℛ^×_𝑆,and the𝐿-algebra isomorphism (2.2) restricts to an𝒪_𝐿,𝑆-algebra isomorphism:

𝑠 ∶ ℛ_𝑆⊗_𝒪

𝐸0,𝑆𝒪_𝐿⁺_,𝑆 = ℛ_𝑆 ⊗_{𝒪𝐸,𝑆}𝒪_𝐿,𝑆 ,→ 𝑀^∼ _𝑑(𝒪_𝐿,𝑆). (2.14) Moreover with these assumptions the involution∗of𝐷 ⊗_𝐸0𝐿⁺restricts to the in- volution∗ofℛ_𝑆⊗_𝒪

𝐸0,𝑆𝒪_𝐿+,𝑆which, in turn, extends to the involution𝑋 ↦ 𝑋^∗ ∶=

𝑋^𝑇𝑟of𝑀_𝑑(𝒪_𝐿,𝑆).

Proof. Follows immediately from Lemma2.1and its proof.

3. Weil pairing of an abelian variety of type IV

Let𝑇(𝐴) = 𝐻₁(𝐴(ℂ), ℤ) and𝑉(𝐴) = 𝑇(𝐴) ⊗_ℤℚ. The polarization on𝐴 induces a non-degenerate alternatingℤ-bilinear form, the Riemann form of𝐴:

𝜅 ∶ 𝑇(𝐴) × 𝑇(𝐴) → ℤ. (3.1)

Let𝜅⁰ ∶= 𝜅 ⊗_ℤℚ ∶ 𝑉(𝐴) × 𝑉(𝐴) → ℚ. Then for all𝑣₁, 𝑣₂ ∈ 𝑉(𝐴)and 𝑥 ∈ 𝐷we have:

𝜅⁰(𝑥𝑣₁, 𝑣₂) = 𝜅⁰(𝑣₁, 𝑥^′𝑣₂). (3.2)

There exists a unique𝐸-bilinear form (𝐸acts on factor𝑉(𝐴)on the right by complex conjugation):

𝜙⁰∶ 𝑉(𝐴) × 𝑉(𝐴) → 𝐸 (3.3)

with𝜅⁰(𝑣₁, 𝑣₂) = 𝑇𝑟_𝐸∕ℚ(𝑓𝜙⁰(𝑣₁, 𝑣₂))where𝑓 ∈ 𝐸and𝑓 = −𝑓[11, Lemma 4.6].

In addition, it is also proven loc. cit. that𝜙⁰ is𝐸-hermitian. Now let𝑇_𝑆 ∶=

𝑇(𝐴) ⊗_𝒪⁰

𝐸𝒪_𝐸,𝑆and𝑉_𝑆 ∶= 𝑇_𝑆⊗_{𝒪𝐸,𝑆}𝐸.Observe that:

𝑉(𝐴) = 𝑇(𝐴) ⊗_ℤℚ = 𝑇(𝐴) ⊗_𝒪⁰

𝐸 𝒪⁰_𝐸⊗_ℤℚ = 𝑇(𝐴) ⊗_𝒪⁰

𝐸𝐸

= 𝑇(𝐴) ⊗_𝒪⁰

𝐸𝒪_𝐸,𝑆⊗_{𝒪𝐸,𝑆}𝐸 = 𝑇_𝑆⊗_{𝒪𝐸,𝑆} 𝐸 = 𝑉_𝑆.

We can enlarge the set𝑆from previous section, if necessary, so that𝑓 ∈ 𝒪^×_𝐸,𝑆 and the𝐸-hermitian form (3.3) restricts to the following𝒪_𝐸,𝑆-hermitian form

𝜙_𝑆 ∶ 𝑇_𝑆× 𝑇_𝑆 → 𝒪_𝐸,𝑆 (3.4)

such that𝜅_𝑆(𝑣₁, 𝑣₂) = 𝑇𝑟_𝐸∕ℚ(𝑓𝜙_𝑆(𝑣₁, 𝑣₂))where𝜅_𝑆 ∶= 𝜅 ⊗_ℤℤ_𝑆.Observe that 𝜙⁰= 𝜙_𝑆⊗_{𝒪𝐸,𝑆} 𝐸.

Recall that we put 𝐸_𝑙 ∶= 𝐸 ⊗_ℚ ℚ_𝑙 and 𝒪_𝐸𝑙 ∶= 𝒪_𝐸 ⊗_ℤ ℤ_𝑙. Note that 𝐸_𝑙 = 𝒪_𝐸

𝑙⊗_ℤ

𝑙 ℚ_𝑙.

From now on till the end of this paper we assume that𝑙 ∉ 𝑆. Then𝒪_𝐸

𝑙 = 𝒪_𝐸,𝑆⊗_ℤ

𝑆ℤ_𝑙.We can naturally extend the action of complex conju- gation on𝐸to the action on rings𝐸_𝑙and𝒪_𝐸𝑙 imposing trivial action onℚ_𝑙.The action will be denoted in the same way as complex conjugation i.e. the action on𝑥 ∈ 𝐸_𝑙 will be denoted𝑥.Observe that𝑓 ∈ 𝒪^×_𝐸

𝑙.By [3, Lemma 3.1] and the idea of the proof of [11, Lemma 4.6] there is a unique 𝒪_𝐸

𝑙-bilinear form (𝒪_𝐸

𝑙

acts on factor𝑇_𝑙(𝐴)on the right by complex conjugation)

𝜙_𝑙 ∶ 𝑇_𝑙(𝐴) × 𝑇_𝑙(𝐴) → 𝒪_𝐸𝑙 (3.5) such that theℤ_𝑙-bilinear form𝜅_𝑙 ∶= 𝜅 ⊗_ℤℤ_𝑙 ∶

𝜅_𝑙 ∶ 𝑇_𝑙(𝐴) × 𝑇_𝑙(𝐴) → ℤ_𝑙 (3.6) has the following property:

𝜅_𝑙(𝑣₁, 𝑣₂) = 𝑇𝑟_{𝐸𝑙∕ℚ𝑙}(𝑓𝜙_𝑙(𝑣₁, 𝑣₂)). (3.7) We can also prove as in loc.cit. that𝜙_𝑙is𝒪_𝐸𝑙-hermitian.

Lemma 3.1. There is the following isomorphism of𝒪_𝐸𝑙-hermitian forms:

𝜙_𝑙 = 𝜙_𝑆⊗_{𝒪𝐸,𝑆}𝒪_𝐸𝑙. (3.8) Proof. There is the following equality inℤ_𝑙 ∶

𝜅_𝑆(𝑢₁, 𝑢₂) ⊗_ℤ𝑆1 = 𝑇𝑟_𝐸∕ℚ(𝑓𝜙_𝑆(𝑢₁, 𝑢₂)) ⊗_ℤ𝑆1 (3.9)

= 𝑇𝑟_{𝐸𝑙∕ℚ𝑙}(𝑓 𝜙_𝑆⊗_{𝒪𝐸,𝑆}𝒪_𝐸𝑙(𝑢₁⊗_{𝒪𝐸,𝑆}1, 𝑢₂⊗_{𝒪𝐸,𝑆} 1)).

Since𝜅_𝑆⊗_ℤ𝑆 ℤ_𝑙 = 𝜅_𝑙and𝒪_𝐸𝑙 = 𝒪_𝐸,𝑆⊗_ℤ𝑆ℤ_𝑙, we obtain by (3.9) the following equality inℤ_𝑙for all𝑢₁, 𝑢₂ ∈ 𝑇_𝑆and𝛼₁, 𝛼₂ ∈ 𝒪_𝐸𝑙 ∶

𝜅_𝑙(𝑢₁⊗ 𝛼₁, 𝑢₂⊗ 𝛼₂) = 𝑇𝑟_{𝐸𝑙∕ℚ𝑙}(𝑓 𝜙_𝑆⊗_{𝒪𝐸,𝑆}𝒪_𝐸𝑙(𝑢₁⊗_{𝒪𝐸,𝑆}𝛼₁, 𝑢₂⊗_{𝒪𝐸,𝑆}𝛼₂)). (3.10) Observe that

𝑇_𝑙(𝐴) = 𝑇(𝐴) ⊗_ℤℤ_𝑙 = 𝑇(𝐴) ⊗_𝒪⁰

𝐸𝒪⁰_𝐸⊗_ℤℤ_𝑙 = 𝑇(𝐴) ⊗_𝒪⁰

𝐸𝒪_𝐸,𝑆⊗_ℤ𝑆 ℤ_𝑙 =

= 𝑇_𝑆 ⊗_{𝒪𝐸,𝑆}𝒪_𝐸,𝑆⊗_ℤ𝑆ℤ_𝑙 = 𝑇_𝑆⊗_{𝒪𝐸,𝑆}𝒪_𝐸𝑙. (3.11) Hence by (3.10), for all𝑣₁, 𝑣₂∈ 𝑇_𝑙(𝐴)we obtain:

𝜅_𝑙(𝑣₁, 𝑣₂) = 𝑇𝑟_{𝐸𝑙∕ℚ𝑙}(𝑓 𝜙_𝑆⊗_{𝒪𝐸,𝑆} 𝒪_𝐸𝑙(𝑣₁, 𝑣₂)). (3.12) By uniqueness of the form𝜙_𝑙 (3.5) and by equality (3.12), we obtain the equal-

ity (3.8).

Lemmas3.2and3.3below extend [8, Lemma (2.3)] to abelian varieties of type IV.

Lemma 3.2. For all𝑣₁, 𝑣₂ ∈ 𝑉(𝐴), 𝑢₁, 𝑢₂ ∈ 𝑇_𝑆, 𝑥 ∈ 𝐷, 𝑦 ∈ ℛ_𝑆 there are the following equalities:

𝜙⁰(𝑥𝑣₁, 𝑣₂) = 𝜙⁰(𝑣₁, 𝑥^′𝑣₂), 𝜙_𝑆(𝑦𝑢₁, 𝑢₂) = 𝜙_𝑆(𝑢₁, 𝑦^′𝑢₂).

Proof. Fix𝑥 ∈ 𝐷. Consider theℚ-bilinear form𝜅_𝑥(𝑣₁, 𝑣₂) ∶ 𝑉(𝐴)×𝑉(𝐴) → ℚ, defined as follows:

𝜅_𝑥(𝑣₁, 𝑣₂) ∶= 𝜅⁰(𝑥𝑣₁, 𝑣₂) = 𝜅⁰(𝑣₁, 𝑥^′𝑣₂).

Consider two𝐸-bilinear forms𝜙⁰₁, 𝜙₂⁰∶ 𝑉(𝐴) × 𝑉(𝐴) → 𝐸 ∶

𝜙⁰₁(𝑣₁, 𝑣₂) ∶= 𝜙⁰(𝑥𝑣₁, 𝑣₂) and 𝜙₂⁰(𝑣₁, 𝑣₂) ∶= 𝜙⁰(𝑣₁, 𝑥^′𝑣₂).

Recall that

𝜅⁰(𝑥𝑣₁, 𝑣₂) = 𝑇𝑟_𝐸∕ℚ(𝑓 𝜙⁰(𝑥𝑣₁, 𝑣₂)) and

𝜅⁰(𝑣₁, 𝑥^′𝑣₂) = 𝑇𝑟_𝐸∕ℚ(𝑓 𝜙⁰(𝑣₁, 𝑥^′𝑣₂)), where𝑓 ∈ 𝐸. Hence

𝜅_𝑥(𝑣₁, 𝑣₂) = 𝑇𝑟_𝐸∕ℚ(𝑓 𝜙⁰₁(𝑣₁, 𝑣₂)) = 𝑇𝑟_𝐸∕ℚ(𝑓 𝜙⁰₂(𝑣₁, 𝑣₂)).

We have𝜙⁰₁= 𝜙⁰₂by [11, Lemma 4.6]. So the first equality follows. Fix𝑦 ∈ ℛ_𝑆. Consider two𝒪_𝐸,𝑆-bilinear forms𝜙₁, 𝜙₂∶ 𝑇_𝑆× 𝑇_𝑆 → 𝒪_𝐸,𝑆defined as follows:

𝜙₁(𝑢₁, 𝑢₂) ∶= 𝜙_𝑆(𝑦𝑢₁, 𝑢₂) and 𝜙₂(𝑢₁, 𝑢₂) ∶= 𝜙_𝑆(𝑢₁, 𝑦^′𝑢₂).

Observe that bilinear forms𝜙_𝑖are restrictions to𝑇_𝑆× 𝑇_𝑆of𝐸-bilinear forms𝜙⁰_𝑖 with𝑦in place of𝑥.Hence, the second equality of the lemma follows from the

first.

Lemma 3.3. For all𝑣₁, 𝑣₂∈ 𝑇_𝑙(𝐴)and𝑔 ∈ 𝐺_𝐹 we have the following equality:

𝜙_𝑙(𝑔𝑣₁, 𝑔𝑣₂) = 𝜒_𝑐(𝑔)𝜙_𝑙(𝑣₁, 𝑣₂).

Here𝜒_𝑐is the cyclotomic character𝜒_𝑐∶ 𝐺_𝐹 → ℤ_𝑙.

Proof. By Galois equivariance of the Weil pairing for all𝑣₁, 𝑣₂∈ 𝑇_𝑙(𝐴)and all 𝑔 ∈ 𝐺_𝐹 we have:

𝜅_𝑙(𝑔𝑣₁, 𝑔𝑣₂) = 𝜒_𝑐(𝑔) 𝜅_𝑙(𝑣₁, 𝑣₂). (3.13) Fix𝑔 ∈ 𝐺_𝐹 and considerℤ_𝑙-bilinear form: 𝜅_𝑔(𝑣₁, 𝑣₂) ∶ 𝑇_𝑙(𝐴) × 𝑇_𝑙(𝐴) → ℤ_𝑙 defined as follows:

𝜅_𝑔(𝑣₁, 𝑣₂) ∶= 𝜅_𝑙(𝑔𝑣₁, 𝑔𝑣₂) = 𝜒_𝑐(𝑔) 𝜅_𝑙(𝑣₁, 𝑣₂).

Consider two𝒪_𝐸𝑙-bilinear forms:𝜙_𝑙¹, 𝜙²_𝑙 ∶ 𝑇_𝑙(𝐴) × 𝑇_𝑙(𝐴) → 𝒪_𝐸𝑙 ∶ 𝜙¹_𝑙(𝑣₁, 𝑣₂) ∶= 𝜙_𝑙(𝑔𝑣₁, 𝑔𝑣₂) and 𝜙²_𝑙(𝑣₁, 𝑣₂) ∶= 𝜒_𝑐(𝑔)𝜙_𝑙(𝑣₁, 𝑣₂).

By (3.7) we obtain

𝜅_𝑔(𝑣₁, 𝑣₂) = 𝑇𝑟_𝐸

𝑙∕ℚ_𝑙(𝑓𝜙¹_𝑙(𝑣₁, 𝑣₂)) = 𝑇𝑟_𝐸

𝑙∕ℚ_𝑙(𝑓𝜙²_𝑙(𝑣₁, 𝑣₂)) for𝑓 ∈ 𝒪^×_𝐸

𝑙. Hence, we obtain𝜙¹_𝑙 = 𝜙_𝑙²by [11, Lemma 4.6].

Now define the following𝒪_𝐸,𝑆-hermitian form

𝜓_𝑆 ∶ 𝑇_𝑆 × 𝑇_𝑆 → 𝒪_𝐸,𝑆, (3.14) 𝜓_𝑆(𝑣₁, 𝑣₂) = 𝜙_𝑆(𝛾⁻¹𝑣₁, 𝑣₂).

Let

𝜓⁰∶= 𝜓_𝑆⊗_{𝒪𝐸,𝑆} 𝐸 ∶ 𝑉 × 𝑉 → 𝐸. (3.15) Because the form (3.1) is non-degenerate, the forms𝜙, 𝜙⁰, 𝜓 and𝜓⁰ are also non-degenerate.

Lemma 3.4. For every𝑥 ∈ ℛ_𝑆and all𝑣₁, 𝑣₂∈ 𝑇_𝑆we have:

𝜓_𝑆(𝑥𝑣₁, 𝑣₂) = 𝜓_𝑆(𝑣₁, 𝑥^∗𝑣₂), where, as defined in previous section,𝑥^∗= 𝛾𝑥^′𝛾⁻¹.

Proof. Recall that𝛾^′ = 𝛾and let𝑥 ∈ ℛ_𝑆.We obtain the following equality for all𝑣₁, 𝑣₂∈ 𝑇_𝑆from the property of Rosati involution, Lemma3.2, the definition of S and the fact that𝜙_𝑆and𝜓_𝑆are𝒪_𝐸,𝑆-hermitian forms.

𝜓_𝑆(𝑥𝑣₁, 𝑣₂) = 𝜙_𝑆(𝛾⁻¹𝑥𝑣₁, 𝑣₂) = 𝜙_𝑆(𝑣₁, 𝑥^′𝛾⁻¹𝑣₂) = 𝜙_𝑆(𝑣₁, 𝛾⁻¹𝛾𝑥^′𝛾⁻¹𝑣₂)

= 𝜙_𝑆(𝑣₁, 𝛾⁻¹𝑥^∗𝑣₂) = 𝜓_𝑆(𝑣₁, 𝑥^∗𝑣₂).

It follows from Lemma2.1that the involution∗induced on𝐷_𝐿∶= 𝐷 ⊗_𝐸𝐿 ≅ 𝑀_𝑑(𝐿)from𝐷is of the form𝐵^∗ = 𝐵^𝑇𝑟for each𝐵 ∈ 𝑀_𝑑(𝐿). Consider the com- plex conjugation𝜏 ∈ Gal(𝐿∕𝐸₀). Take a prime number𝑙and𝜆₀|𝑙in𝒪_𝐸0 such thatFrob_𝜔∕𝜆0 = 𝜏for a prime ideal𝜔 ∈ Spec(𝒪_𝐿). Let𝜆 = 𝒪_𝐸∩ 𝜔be the prime ideal in𝒪_𝐸 below𝜔and over𝜆₀. Because the order of𝜏is2inGal(𝐿∕𝐸₀)and 𝜏 ∈ Gal(𝐸∕𝐸₀)is also of order2, hence𝜆is inert over𝜆₀and𝜆splits completely in𝒪_𝐿. There are infinitely many such primes𝜆by Chebotarev’s theorem. Then we have

[𝐿_𝜔 ∶ 𝐸_𝜆] = 1 and 𝒪_𝜆 = 𝒪_𝜔. (3.16) Put

𝑇_𝜆(𝐴) ∶= 𝑇_𝑆⊗_{𝒪𝐸,𝑆} 𝒪_𝜆, 𝑉_𝜆(𝐴) ∶= 𝑇_𝜆(𝐴) ⊗_𝒪𝜆𝐸_𝜆, 𝐴[𝜆] ∶= 𝑇_𝜆(𝐴)∕𝜆𝑇_𝜆(𝐴).

(3.17) Note that𝐴[𝜆]is a𝑘_𝜆[𝐺_𝐹]-module. Define a𝜆-adic hermitian form as follows:

𝜙_𝜆 ∶= 𝜙_𝑆⊗_{𝒪𝐸,𝑆} 𝒪_𝜆 ∶ 𝑇_𝜆(𝐴) × 𝑇_𝜆(𝐴) → 𝒪_𝜆. (3.18) Observe that 𝜙_𝜆 = 𝜙_𝑙 ⊗_𝒪

𝐸𝑙 𝒪_𝜆 by Lemma3.8. We also obtain the following hermitian forms:

𝜙⁰_𝜆∶= 𝜙_𝜆⊗_𝒪𝜆𝐸_𝜆 ∶ 𝑉_𝜆(𝐴) × 𝑉_𝜆(𝐴) → 𝐸_𝜆, 𝜙_𝜆 ∶= 𝜙_𝜆⊗_𝒪𝜆𝑘_𝜆 ∶ 𝐴[𝜆] × 𝐴[𝜆] → 𝑘_𝜆. Since𝛾^′= 𝛾, the following forms are also hermitian:

𝜓_𝜆∶ 𝑇_𝜆(𝐴) × 𝑇_𝜆(𝐴) → 𝒪_𝜆, (3.19) 𝜓_𝜆(𝑣₁, 𝑣₂) = 𝜙_𝜆(𝛾⁻¹𝑣₁, 𝑣₂),

𝜓⁰_𝜆 ∶= 𝜓_𝜆⊗_𝒪𝜆𝐸_𝜆 ∶ 𝑉_𝜆(𝐴) × 𝑉_𝜆(𝐴) → 𝐸_𝜆, (3.20) 𝜓_𝜆 ∶= 𝜓_𝜆⊗_𝒪𝜆𝑘_𝜆 ∶ 𝐴[𝜆] × 𝐴[𝜆] → 𝑘_𝜆. (3.21) Lemma 3.5. Hermitian forms 𝜙_𝜆, 𝜙⁰_𝜆, 𝜙_𝜆, 𝜓_𝜆, 𝜓⁰_𝜆, 𝜓_𝜆 are non-degenerate and𝐺_𝐹-equivariant.

Proof. Since the form𝜅_𝑙(3.6) is non-degenerate, the form𝜙_𝑙 (3.5) is also non- degenerate by property (3.7). Consider the bilinear forms:

𝜅_𝑙 ∶= 𝜅_𝑙⊗_ℤ𝑙 ℤ∕𝑙 ∶ 𝐴[𝑙] × 𝐴[𝑙] → ℤ∕𝑙, 𝜙_𝑙 ∶= 𝜙_𝑙⊗_𝒪

𝐸𝑙 𝒪_𝐸𝑙∕𝑙 ∶ 𝐴[𝑙] × 𝐴[𝑙] → 𝒪_𝐸𝑙∕𝑙 related by the following equality

𝜅_𝑙(𝑣₁, 𝑣₂) = 𝑇𝑟_{𝐸𝑙∕ℚ𝑙}(𝑓𝜙_𝑙(𝑣₁, 𝑣₂)). (3.22) where𝑓 ∈ 𝒪^×_𝐸

𝑙. Because𝑙does not divide the polarisation of𝐴, then𝜅_𝑙(𝑣₁, 𝑣₂) is non-degenerate. Hence,𝜙_𝑙(𝑣₁, 𝑣₂)is non-degenerate by (3.22). By [3, Lemma 3.2], forms 𝜙_𝜆, 𝜙⁰_𝜆, 𝜙_𝜆are non-degenerate. Hence, it is obvious that the forms

𝜓_𝜆, 𝜓⁰_𝜆, 𝜓_𝜆are non-degenerate. It follows immediately from Lemma3.3that 𝜙_𝜆, 𝜙_𝜆⁰, 𝜙_𝜆 are𝐺_𝐹-equivariant. By definition of𝜓_𝜆, it follows that the forms 𝜓_𝜆, 𝜓⁰_𝜆, 𝜓_𝜆are𝐺_𝐹-equivariant, because𝐺_𝐹commutes withEnd_𝐹(𝐴).

Observe that we have the following isomorphism

𝐷_𝜆∶= 𝐷 ⊗_𝐸𝐸_𝜆≅ 𝑀_𝑑(𝐸_𝜆). (3.23) Indeed, by (3.16) we have𝐷_𝜆= 𝐷⊗_𝐸𝐸_𝜆 = 𝐷⊗_𝐸𝐿_𝜔= 𝐷⊗_𝐸𝐿⊗_𝐿𝐿_𝜔= 𝑀_𝑑(𝐿_𝜔) = 𝑀_𝑑(𝐸_𝜆). Then (3.23) induces the following isomorphism of𝒪_𝜆-modules

ℛ_𝜆∶= ℛ_𝑆⊗_{𝒪𝐸,𝑆}𝒪_𝜆 ≅ 𝑀_𝑑(𝒪_𝜆). (3.24) By (1.1), we have

ℛ ⊗_ℤℤ_𝑙 = ℛ ⊗_𝒪⁰

𝐸𝒪_𝐸

𝑙 = ℛ_𝑆⊗_𝒪

𝐸,𝑆 𝒪_𝐸

𝑙 =∏

𝜆|𝑙

ℛ_𝜆. (3.25) On the other hand, by [12, Satz 4]:

ℛ ⊗_ℤℤ_𝑙 ,→ End^∼ _{ℤ𝑙[𝐺𝐹]}(𝑇_𝑙(𝐴)). (3.26) By (1.2), (3.25) and (3.26), we obtain the following isomorphism of𝒪_𝜆-algebras.

ℛ_𝜆 ,→ End^∼ _𝒪

𝜆[𝐺_𝐹](𝑇_𝜆(𝐴)). (3.27) Finally, (3.24) and (3.27) give the following isomorphism of𝒪_𝜆-algebras:

End_{𝒪𝜆[𝐺𝐹]}(𝑇_𝜆(𝐴)),→ 𝑀^∼ _𝑑(𝒪_𝜆). (3.28) Remark 3.6. Since𝜆|𝜆₀is unramified and inert, we have

Gal(𝐸∕𝐸₀) ≅ Gal(𝐸_𝜆∕𝐸_0,𝜆0) ≅ Gal(𝑘_𝜆∕𝑘_𝜆0).

Hence, the elementFrob_𝜆∕𝜆0 = 𝜏 ∈ Gal(𝐸∕𝐸₀)can be considered as an element inGal(𝐸_𝜆∕𝐸_0,𝜆0). Thus if a matrix𝐵 ∈ 𝑀_𝑑(𝐸)is considered as an element of 𝑀_𝑑(𝐸_𝜆),𝜏acts on𝐵viaFrob_𝜆∕𝜆0 and we will denote𝐵 ∶= Frob_𝜆∕𝜆0(𝐵).

Proposition 3.7. (i) For every𝑣₁, 𝑣₂∈ 𝑇_𝜆(𝐴)and𝐵 ∈ ℛ_𝜆, we have 𝜓_𝜆(𝐵𝑣₁, 𝑣₂) = 𝜓_𝜆(𝑣₁, 𝐵^𝑇𝑟𝑣₂).

(ii) For every𝑣₁, 𝑣₂ ∈ 𝑉_𝜆(𝐴)and𝐵 ∈ 𝐷_𝜆, we have 𝜓⁰_𝜆(𝐵𝑣₁, 𝑣₂) = 𝜓⁰_𝜆(𝑣₁, 𝐵^𝑇𝑟𝑣₂).

(iii) For every𝑣₁, 𝑣₂ ∈ 𝐴[𝜆]and𝐵 ∈ ℛ_𝜆⊗_𝒪𝜆𝑘_𝜆 ≅ 𝑀_𝑑(𝑘_𝜆), we have 𝜓_𝜆(𝐵𝑣₁, 𝑣₂) = 𝜓_𝜆(𝑣₁, 𝐵^𝑇𝑟𝑣₂).

Proof. It follows from Lemmas2.1,3.4and the isomorphism (3.24).

Definition 3.8. Let𝒫be the set of prime numbers𝑙 ∉ 𝑆such that there is𝜆₀|𝑙 in𝒪_𝐸

0and𝜆inert over𝜆₀in𝒪_𝐸and𝜆splits completely in𝒪_𝐿(see the discussion below Lemma3.4).

Remark 3.9. Observe that the set𝒫has a positive Dirichlet’s density because of Chebotarev’s theorem. Our main results Theorems4.3and5.2will be for- mulated for primes𝑙 ∈ 𝒫.

4. Main theorem for𝒅 ≤ 𝟐

Based on results of previous sections, we construct the Tate module decom- position for an abelian variety of type IV when𝑑 = [𝐷 ∶ 𝐸]

1

2 = 2. We observe that we can prove Theorem4.3applying the same idempotents as for types II and III as well as the standard idempotents. This observation is the key for proving our main result for arbitrary degree𝑑 = [𝐷 ∶ 𝐸]

1

2 which will be shortly described in the next section. We also briefly explain, at the beginning of the proof of Theorem4.3, how the construction works for the case𝑑 = 1.

Following [8, p. 91-92], consider the following matrices:

𝑡 = (1 0

0 −1) , 𝑢 = (0 1 1 0) . Consider the idempotent𝑒 = ¹

2(1 + 𝑡) = (1 0

0 0). Define:

𝒳 ∶= 𝑒 𝑇_𝜆(𝐴) and 𝒴 = (1 − 𝑒) 𝑇_𝜆(𝐴),

𝑋 ∶= 𝒳 ⊗_𝒪𝜆𝐸_𝜆, 𝑌 ∶= 𝒴 ⊗_𝒪𝜆𝐸_𝜆, 𝒳 ∶= 𝒳 ⊗_𝒪𝜆𝑘_𝜆, 𝒴 ∶= 𝒴 ⊗_𝒪𝜆𝑘_𝜆. By (3.27), the action ofℛ_𝜆 commutes with the action of𝒪_𝜆[𝐺_𝐹]on𝑇_𝜆(𝐴).

Hence, the equality𝑢 𝑒 𝑢 = (1 − 𝑒)yields a𝒪_𝜆[𝐺_𝐹]- isomorphism between𝒳 and𝒴, a𝐸_𝜆[𝐺_𝐹]- isomorphism between𝑋and𝑌, and a𝑘_𝜆[𝐺_𝐹]- isomorphism between𝒳and𝒴.

Lemma 4.1. [4, Lemma 3.22]Modules𝒳and𝒴are orthogonal with respect to 𝜓_𝜆. Moreover, modules𝑋and𝑌are orthogonal with respect to𝜓⁰_𝜆, and𝒳and𝒴 are orthogonal with respect to𝜓_𝜆.

Proof. Note that𝑡 𝑒 = 𝑒and𝑡(1 − 𝑒) = −(1 − 𝑒). Then for every𝑣₁ ∈ 𝒳and 𝑣₂∈ 𝒴, we obtain𝑡 𝑣₁= 𝑣₁and𝑡 𝑣₂= −𝑣₂. Hence by Proposition3.7, we obtain

𝜓_𝜆(𝑣₁, 𝑣₂) = 𝜓_𝜆(𝑡 𝑣₁, 𝑣₂) = 𝜓_𝜆(𝑣₁, 𝑡^∗𝑣₂) = 𝜓_𝜆(𝑣₁, 𝑡^𝑇𝑟𝑣₂)

= 𝜓_𝜆(𝑣₁, 𝑡 𝑣₂) = 𝜓_𝜆(𝑣₁, −𝑣₂) = −𝜓_𝜆(𝑣₁, 𝑣₂).

Hence,𝜓_𝜆(𝑣₁, 𝑣₂) = 0for every𝑣₁∈ 𝒳and for every𝑣₂∈ 𝒴. The discussion before Lemma4.1gives the following isomorphism of𝒪_𝜆[𝐺_𝐹]- modules

𝑇_𝜆(𝐴) ≅ 𝒳 ⊕ 𝒳. (4.1)

Then by (3.28) we obtain the following isomorphism of𝒪_𝜆-algebras,

𝑀₂(End_{𝒪𝜆[𝐺𝐹]}(𝒳)),→ End^∼ _{𝒪𝜆[𝐺𝐹]}(𝑇_𝜆(𝐴)),→ 𝑀^∼ ₂(𝒪_𝜆). (4.2)