On the Structure of Weil Restrictions of Abelian Varieties

Claus Diem  Niko Naumann

June 10, 2003

Abstract

We give a description of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions of fields. The results are applied to study the isogeny decompositions of Weil restrictions.

2000 Mathematics Subject Classification Primary: 14K15, Secondary: 11G10.

Introduction

For the use of Weil restrictions of abelian varieties in various fields of mathematics but also because of genuine interest in Weil restrictions themselves, it is important to determine the endomorphism rings and the isogeny decompositions. This is what this article provides – at least in two important special cases.

After giving a brief exposé of general facts about Weil restrictions of abelian varieties in the first section, we study Weil restrictions with respect to extensions of finite fields in the second section. Here we determine the endomorphism algebra of a Weil restriction (see Theorem 1) and then show that under rather general assumptions, the Weil restriction is simple over the base-field (see Theorem 2).

In the third section, we deal with the following situation: K|k is an arbitrary finite Galois extension of fields, A an abelian variety over k, W the Weil restriction of A_{K} with respect to K|k. We describe the endomorphism ring of W as a skew group ring over End(A_{K}) (see Theorem 3) and apply this result to study the isogeny decomposition of W over k. In the last subsection the results are applied to give an explicit description of the isogeny decomposition of W in the case of a cyclic field extension; see Theorem 4.

Notation

General

By a ring we mean a ring with unity, and by a ring-homomorphism a homomorphism of rings with unity. If R is a ring and Σ a finite set, then by M_{Σ}(R) we mean the matrix ring over R on the set Σ. For any abelian group G, G^{∘} denotes G ⊗ ℚ. If D is a skew field, we denote its center by Z(D). If k is a field, \overline{k} denotes an algebraic closure. If k is some field and X and Z are k-schemes, we denote the k-morphisms from Z to X by X(Z).

Let k be a field. By a homomorphism between abelian k-varieties we mean a morphism of k-schemes which preserves the group structure (other authors might call this a k-homomorphism or a k-morphism of abelian varieties). Analogous definitions apply to isogenies and endomorphisms. The group of homomorphisms between two abelian k-varieties A and B is denoted by Hom(A, B) and the ring of endomorphisms of an abelian k-variety A by End(A). Following this terminology, we use the notion of a simple abelian k-variety where other authors might speak of a k-simple abelian k-variety. If two abelian k-varieties A and B are isogenous, we write A ∼ B.

If we are given an extension of fields K|k, we denote k-schemes by X, Y etc. and K-schemes by X′, Y′ etc. (or by X_{K}, Y_{K} etc. if they are induced by base-change K|k).

We denote the dual abelian variety of an abelian k-variety A by Â. For an invertible sheaf ℒ on A, ϕ_{ℒ} : A ⟶ Â denotes the corresponding homomorphism; c.f. [7, §6]. Following [6], a polarization φ of A is a homomorphism A ⟶ Â such that φ ⊗_{k} id\overline{k} = ϕ_{ℒ} : A\overline{k} ⟶ Â\overline{k} for some ample invertible sheaf on A\overline{k}.

Galois twists

Let K|k be a Galois extension of fields with Galois group G. Then the elements of G induce automorphisms of the Spec(k)-scheme Spec(K) – we obtain in this way an anti-isomorphism G ⟶ Aut_{Spec(k)}(Spec(K)).

We identify the opposite group G^{opp} with Aut_{Spec(k)}(Spec(K)). We will always work with G^{opp} instead of G.

Let X′ be a K-scheme.

For σ ∈ G^{opp}, let σ^{−1}(X′) be the pull-back of X′ via σ : Spec(K) ⟶ Spec(K), i.e. if p_{K} : X′ ⟶ Spec(K) is the structure morphism, σ^{−1}(X′) is X′ considered as K-scheme via σ^{−1} ∘ p_{K}. We denote the canonical isomorphism of k-schemes from σ^{−1}(X′) to X′ also by σ. If Y′ is another K-scheme and α : X′ ⟶ Y′ is a morphism of K-schemes, we obtain by base-change a morphism of K-schemes σ^{−1}(α) = σ^{−1}ασ : σ^{−1}(X′) ⟶ σ^{−1}(Y′).

On the Structure of Weil Restrictions of Abelian Varieties 3

If X′ is an abelian K-variety, by pull-back σ^{−1}(X′) also has the structure of an abelian K-variety.

Frobenius morphisms

Let q be a power of a prime number, k the finite field with q elements, let A be an abelian k-variety. The Frobenius endomorphism π_{k} of A is defined by the identity on the underlying topological space and by f ↦ f^{q} on the structure-sheaf 𝒪_{A}. As the name indicates, π_{k} is an endomorphism of the abelian k-variety A.

Now let K|k be an algebraic extension of fields. We identify the Galois group Gal(K|k) with its dual. The Frobenius automorphism of K|k (or of Spec(K) ⟶ Spec(k)) is denoted by σ_{K|k}. If K = \overline{k}, we write σ_{k} instead of σ_{\overline{k}|k}.

Let A′ be an abelian K-variety. As stated above, we have a canonical isomorphism of k-schemes σ_{K|k} : σ_{K|k}^{−1}(A′) ⟶ A′. The relative Frobenius homomorphism (with respect to k) π_{k} : A′ ⟶ σ_{K|k}^{−1}(A′) is a homomorphism of abelian K-varieties which is defined as follows: Let F_{k} be the morphism of the k-scheme A′ to itself which is the identity on the underlying topological space and it is given by f ↦ f^{q} on the structure-sheaf 𝒪_{A′}. Then π_{k} := σ_{K|k}^{−1} ∘ F_{k} : A′ ⟶ σ_{K|k}^{−1}(A′).

1 Definitions and first results

1.1 Definition of the Weil restriction

Let K|k be a finite Galois extension. Let A′ an abelian K-variety. It is well-known that the functor

Z ↦ A′(Z ⊗_{k} K)

from the category of k-schemes to the category of abelian groups is representable by an abelian k-variety; for a construction via Galois theory see Subsection 1.2, for a construction via “restriction of scalars” see [1, 7.6]. (The representatility of the functor by an abelian variety holds more generally for a finite separable extension of fields, but we restrict ourselves to the Galois-case is this article.) A representing object will be denoted Res_{k}^{K}(A′) and will be called the Weil restriction of A′ with respect to K|k. The universal element u ∈ A′(Res_{k}^{K}(A′) ⊗_{k} K) maps the zero of Res_{k}^{K}(A′) ⊗_{k} K to the zero of A′ and thus is a homomorphism of abelian K-varieties.

Now, Res_{k}^{K}(A′) with u is also a representing object for the functor B ↦ Hom(B_{K}, A′) from the category of abelian k-varieties to the category of abelian groups as well as for the functor B ↦ Hom^{∘}(B_{K}, A′) from the category of abelian k-varieties up to isogeny to the category of ℚ-vector spaces.

1.2 Construction of the Weil restriction

Let us recall the construction of Res_{k}^{K}(A′) via Galois theory.

Let W′ be the following product of Galois-conjugates of A′:

W′ := ∏_{σ∈G^{opp}} σ^{−1}(A′)  (1)

Let p_{σ} : W′ ⟶ σ^{−1}(A′) be the projections, let Aut_{k}(W′) be the group of automorphisms of the k-scheme W′.

We define a Galois operation on W′ by G^{opp} ⟶ Aut_{k}(W′), τ ↦ τ̃ where τ̃ = (τ p_{στ})_{σ∈G^{opp}}. Since W′ is projective, the quotient W := W′/G under this operation exists and is projective. We have W′ ≃ W_{K}.

Fix some k-scheme Z. We have a Galois operation on W′(Z ⊗_{k} K). If τ ∈ G^{opp} and P = (P_{σ})_{σ∈G^{opp}} ∈ W′(Z ⊗_{k} K), then τ((P_{σ})_{σ∈G^{opp}}) = (τ(P_{στ}))_{σ∈G^{opp}}. It follows that P ↦ (σ^{−1}(P))_{σ∈G^{opp}} is a bijection between the Z ⊗_{k} K-valued points of A′ and the Galois-invariant Z ⊗_{k} K-valued points of W′. On the other hand, by Galois theory, the Galois-invariant Z ⊗_{k} K-valued points of W′ are in bijection with the Z-valued points of W. Both bijections are natural in Z.

It follows that W = W′/G with universal element u := p_{id} represents the functor Z ↦ A′(Z ⊗_{k} K) from the category of k-schemes to the category of sets. Via the group laws on these sets, one defines a group law on W, and with this group law, W is an abelian variety. By construction, the neutral element and the addition law of W coincide after base-change with the neutral element and the addition law of the product of Galois-conjugates in (1). Moreover, the universal element u = p_{id} is a homomorphism of abelian K-varieties.

From the Galois-operation of W′, we obtain

τ^{−1}(p_{σ}) = p_{στ} , especially τ^{−1}(u) = p_{τ} .  (2)

1.3 The functor “restriction of scalars”

The assignment A′ ↦ Res_{k}^{K}(A′) defines a covariant additive functor Res_{k}^{K} from the category of abelian K-varieties (up to isogeny) to the category of abelian k-varieties (up to isogeny). This functor is called “restriction of the field of definition” or “restriction of scalars” or “norm functor”; cf. [5].

For any abelian K-variety A′, Res_{k}^{K} gives a ring-homomorphism from End(A′) to End(Res_{k}^{K}(A′)) and from End^{∘}(A′) to End^{∘}(Res_{k}^{K}(A′)).

Let A′, B′ be abelian K-varieties. Then

Hom(Res_{k}^{K}(A′)_{K}, Res_{k}^{K}(B′)_{K}) ≃ ⨁_{σ,ν∈G^{opp}} Hom(ν^{−1}(A′), σ^{−1}(B′)); (3)

see equations (1) and (5).

Let α : A′ ⟶ B′ be a homomorphism. Then under (3), Res_{k}^{K}(α)⊗_{k} id_{K} is given by the diagonal “matrix”

(σ^{−1}(α)δ_{σ,ν})_{σ,ν∈G^{opp}} ∈ ⨁_{σ,ν∈G^{opp}} Hom(ν^{−1}(A′), σ^{−1}(B′)),

where δ_{σ,ν} is the “Kronecker delta”. If α : A′ ⟶ B′ is an isogeny, then Res_{k}^{K}(α) : Res_{k}^{K}(A′) ⟶ Res_{k}^{K}(B′) is an isogeny of degree (deg(α))^{[K:k]}.

1.4 The Weil restriction of the dual abelian variety

The Weil restriction of the dual abelian variety is functorially isomorphic to the dual abelian variety of the Weil restriction. This can be seen as follows.

Let W := Res_{k}^{K}(A′).

Let Z be some k-scheme, ℒ some invertible sheaf on A′ ×_{K} Z_{K}, algebraically equivalent to zero. Now consider the invertible sheaf

ℒ_{W_{K}} := ⨂_{σ} p_{σ}^{∗}σ^{∗}(ℒ) = ⨂_{σ} σ^{∗}u^{∗}(ℒ)

on W_{K}. The isomorphism class in Pic(W_{K}×_{K}Z_{K})/Pic(Z_{K}) of this invertible sheaf corresponds to an element in Ŵ_{K}(Z_{K}) which is invariant under the Galois-operation and thus defines an element in Ŵ(Z).

We obtain in this way a homomorphism Â′(Z_{K}) ⟶ { \atop Res_{k}^{K}(A′)}(Z) which is functorial in Z. We thus have a homomorphism Res(Â′) ⟶ { \atop Res_{k}^{K}(A′)}.

After base-change K|k, this homomorphism becomes the canonical isomorphism

∏_{σ∈G^{opp}} σ^{−1}(Â′) ⟶ { \atop ∏_{σ∈G^{opp}} σ^{−1}(A′)},

thus it is an isomorphism. This isomorphism Res(Â′) ⟶ { \atop Res_{k}^{K}(A′)} is functorial in A′ as can for example easily be seen after base-change K|k. We thus have:

Proposition 1 For abelian K-varieties A′, Res_{k}^{K}(Â′) is functorially isomorphic to { \atop Res_{k}^{K}(A′)}.

1.5 Weil restrictions of polarized abelian varieties

Let K|k be a finite Galois field extension, A′ an abelian K-variety, Â′ the dual abelian variety.

Let φ : A′ ⟶ Â′ be a polarization of A′, defined by an ample invertible sheaf ℒ on A′ ; \overline{K}, i.e. φ ⊗_{K} id\overline{K} = ϕ_{ℒ} : A′ ; \overline{K} ⟶ Â′\overline{K}. As stated in Subsection 1.3, this induces an isogeny

Res_{k}^{K}(φ) : Res_{k}^{K}(A′) ⟶ Res_{k}^{K}(Â′) ≃ { \atop Res_{k}^{K}(A′)}.

We show now that this homomorphism is again a polarization.

Let σ ∈ G^{opp}. We regard σ^{−1}(Â′) as the dual abelian variety of σ^{−1}(A′). Let σ′ be a Spec(\overline{K})-automorphism with π ∘ σ′ = σ for the natural map

π : Spec(\overline{K}) → Spec(K). Then

σ^{−1}(φ) ⊗_{K} id\overline{K} = σ′^{−1}(ϕ_{ℒ}) = ϕ_{σ′^{∗}(ℒ)} .

Here, the first equation is obvious by the definition of σ′ and the second equation is a general fact for all polarizations on abelian varieties. It can be checked rather easily on \overline{K}-valued points.

After base-change, we get

Res_{k}^{K}(φ) ⊗_{k} id_{K} = (σ^{−1}(φ) ∘ p_{σ})_{σ∈G^{opp}} : ∏_{σ∈G^{opp}} σ^{−1}(A′) ⟶ ∏_{σ∈G^{opp}} σ^{−1}(Â′).

This is a product polarization defined by the ample invertible sheaf

ℒ_{W\overline{k}} := ⨂_{σ}(p_{σ} ⊗_{K} id\overline{K})^{∗}σ′^{∗}(ℒ)  (4)

on W\overline{k}.

If one starts with an ample invertible sheaf ℒ on A′, then analogously to (4), one defines an ample invertible sheaf ℒ_{W_{K}} on W_{K}. The class of this sheaf in the Picard group is invariant under the operation of Gal(K|k) and thus defines an ample invertible sheaf on W (because the Picard functor of an abelian variety is representable) – alternatively, one can also define explicitely a descent-datum on ℒ_{W_{K}}.

Proposition 2 Let K|k be a finite Galois field extension, A′ an abelian K-variety. If φ is a polarization on A′ (defined by a sheaf on A′), then Res_{k}^{K}(φ) is a polarization on Res_{k}^{K}(A′) (defined by a sheaf on Res_{k}^{K}(A′)). Furthermore deg(Res_{k}^{K}(φ)) = (deg(φ))^{[K:k]}.

Thus “restriction of scalars” is a functor from the category of polarized abelian K-varieties (with polarizations defined by sheaves on A′) to the category of polarized abelian k-varieties (with polarizations defined by sheaves on Res_{k}^{K}(A′)) which preserves principal polarizations.

1.6 Appendix to Section 1: Products and the Rosati involution

Let k be a field, let B_{i} for i = 1, …, m and A_{j} for j = 1, …, n be abelian k-varieties. Let A := ∏_{j=1,...,n} A_{j}, B := ∏_{i=1,...,m} B_{i}. Let ι_{j}^{A} : A_{j} ⟶ A be the inclusions and let p_{j}^{A} : A ⟶ A_{j} be the projections. (Similar definitions for B as well as the corresponding dual abelian varieties Â and B̂.) Then

\begin{matrix}Hom(A, B) & ⟶ & ⨁_{i,j} Hom(A_{j}, B_{i}) \\ ψ & ↦ & (p_{i}^{B}ψι_{j}^{A})_{i=1,...,m, j=1,...,n} \end{matrix}  (5)

is an isomorphism. (The same is true for the corresponding groups Hom^{∘}(…, …) of both sides.)

Thus every homomorphism from A to B is uniquely determined by its “matrix”, and conversely, every “matrix” determines a homomorphism. Furthermore, the composition of homomorphisms corresponds to the usual multiplication of matrices.

In particular, under (5), End(A) is isomorphic to the “matrix ring” ⨁_{i,j} Hom(A_{j}, A_{i}).

For later use we want to study how the Rosati involution with respect to a product polarization operates on the “matrices”. It is convenient to generalize the concept of a “Rosati involution” first.

Let X and Y be abelian k-varieties with fixed polarizations φ_{X} : X ⟶ X̂, φ_{Y} : Y ⟶ Ŷ. Then for every ψ ∈ Hom^{∘}(X, Y), we denote φ_{X}^{−1} ψ̂φ_{Y} ∈ Hom^{∘}(Y, X) by ψ′ and call it the Rosati involution of ψ with respect to φ_{X} and φ_{Y}.

Now for i = 1, …, m, j = 1, …, n, let φ_{B_{i}} : B_{i} ⟶ B̂_{i} and φ_{A_{j}} : A_{i} ⟶ Â_{j} be polarizations. Let φ_{A} and φ_{B} be the corresponding product polarizations.

Lemma 3 Let ψ ∈ Hom^{∘}(A, B) be given by the “matrix” (ψ_{i,j})_{i=1,...,m, j=1,...,n}, ψ_{i,j} ∈ Hom^{∘}(A_{j}, B_{i}). Then with respect to φ_{A} and φ_{B}, the Rosati involution of ψ is given by the “matrix” (ψ′_{j,i})_{i=1,...,n, j=1,...,m} with ψ′_{j,i} ∈ Hom^{∘}(B_{j}, A_{i}).

Proof Straightforward calculation. □

2 Results for finite fields

Let K|k be a finite extension of finite fields of degree n. Let A′ be an abelian variety over K, W the Weil restriction of A′ with respect to K|k.

2.1 The endomorphism algebra

We now study the endomorphism algebra and the isogeny decomposition of W over k.

Let π_{k} : A′ ⟶ σ_{K|k}^{−1}(A′) be the relative Frobenius homomorphism with respect to k and let π_{k} : W ⟶ W be the Frobenius endomorphism; cf. “Notation”.

Let π_{K} be the Frobenius endomorphism of A′. Then the image of π_{K} under the ring-homomorphism Res_{k}^{K} equals the endomorphism π_{k}^{n} of W. (In fact, after base-change K|k, Res_{k}^{K}(π_{K}) as well as π_{k}^{n} become equal to the Frobenius endomorphism of W_{K}.) Thus the ring-homomorphism Res_{k}^{K} : End(A′) ⟶ End(W) restricts to an inclusion ℤ[π_{K}] ⟶ End(W), given by π_{K} ↦ π_{k}^{n}. This ring-homomorphism extends to a ring-homomorphism ℤ[π_{K}][X]/(X^{n} − π_{k}) ⟶ End(W), given by X ⟶ π_{k}.

The Frobenius endomorphism π_{k} of W commutes with all endomorphisms of W. Thus by the universal property of the tensor product, the ringhomomorphisms End(A′) ⟶ End(W), λ ↦ Res_{k}^{K}(λ) and ℤ[π_{K}][X]/(X^{n} − π_{K}) ⟶ End(W), X ↦ π_{k} induce a ring-homomorphism

End(A′) ⊗_{ℤ[π_{K}]} ℤ[π_{K}][X]/(X^{n} − π_{K}) ⟶ End(W), λ ↦ Res_{k}^{K}(λ), X ↦ π_{k}.

Theorem 1 Let K|k be an extension of degree n of finite fields. Let A′ be an abelian K-variety, W the Weil restriction of A′ with respect to K|k. Then

End^{∘}(A′) ⊗_{ℚ[π_{K}]}ℚ[π_{K}][X]/(X^{n} − π_{K}) ⟶ End^{∘}(W), λ ↦ Res_{k}^{K}(λ), X ↦ π_{k}

is an isomorphism.

Proof By the defining property of the Weil restriction, as abelian groups,

Hom^{∘}(W, W) ≃ Hom^{∘}(∏_{i=0}^{n−1} σ_{K|k}^{−i}(A′), A′) via a ↦ p_{id} ∘ (a ⊗_{k} id_{K}). (6)

We show that the homomorphism of abelian groups

Hom^{∘}(A′, A′) ⊗_{ℚ[π_{K}]} ℚ[π_{K}][X]/(X^{n} − π_{K}) ⟶ Hom^{∘}(W, W) ≃

Hom^{∘}(∏_{i=0}^{n−1} σ_{K|k}^{−i}(A′), A′) ≃ ⨁_{i=0}^{n−1} Hom^{∘}(σ_{K|k}^{−i}(A′), A′) (7)

is an isomorphism. Since we already know the homomorphism in the theorem to be a ring-homomorphism, this will conclude the proof.

Let σ_{k} ∈ Gal(\overline{k}|k) be the Frobenius automorphism. By base-change, this induces an automorphism σ_{k} of the k-scheme W\overline{k}.

The endomorphism π_{k} : W ⟶ W is uniquely determined by the fact that it operates on \overline{k}-valued points P of W\overline{k} as the inverse of the “arithmetic Frobenius operation”: (π_{k} ⊗_{k} id\overline{k}) ∘ P = σ_{k}^{−1}(P).

Let P = (P_{i})_{i=0}^{n−1} be a \overline{k}-valued point of W\overline{k} ≃ ∏_{i=0}^{n−1} σ_{K|k}^{−i}(A′)\overline{K}. Then σ_{k}^{−1}(P) = (σ_{k}^{−1}(P_{i−1}))_{i=0}^{n−1} (where P_{−1} := P_{n−1}); see Subsection 1.2. Thus (π_{k} ⊗_{k} id\overline{k}) ∘ P = σ_{k}^{−1}(P) = (σ_{k}^{−1}(P_{i−1}))_{i=0}^{n−1} = ((π_{k} ⊗_{k} id\overline{k}) ∘ P_{i−1})_{i=0}^{n−1}.

It follows that under the isomorphism W_{K} ≃ ∏_{i=0}^{n−1} σ_{K|k}^{−i}(A′), the endomorphism π_{k} ⊗_{k} id_{K} of W_{K} is given by the “matrix”

( \begin{matrix}0 & ⋯ & ⋯ & π_{k} \\ π_{k} & 0 & ⋯ & 0 \\ 0 & ... & ... & ⋮ \\ 0 & ... & π_{k} & 0 \end{matrix} ) .

For λ ∈ End^{∘}(A′), Res_{k}^{K}(λ) ⊗_{k} id_{K} is given by the diagonal “matrix”

( \begin{matrix}λ \\ & σ_{K|k}^{−1}(λ) \\ & & . & . & . \\ & & & & & σ_{K|k}^{−(n−1)}(λ) \end{matrix} );

see Subsection 1.3. Let x denote the image of X in ℚ[π_{K}][X]/(X^{n} − π_{K}). Let λ_{1}x+λ_{2}x^{2}+⋯+λ_{n}x^{n} ∈ Hom^{∘}(A′, A′)⊗_{ℚ[π_{K}]}ℚ[π_{K}][X]/(X^{n}−π_{K}) where λ_{i} ∈ End^{∘}(A′). Such an element is mapped under the homomorphism of the theorem to an endomorphism of W which is represented by the “matrix”

( \begin{matrix}λ_{n} π_{k}^{n} & λ_{n−1} π_{k}^{n−1} & ⋯ & λ_{2} π_{k}^{2} & λ_{1} π_{k} \\ σ_{K|k}^{−1}(λ_{1}) π_{k} & σ_{K|k}^{−1}(λ_{n}) π_{k}^{n} & & σ_{K|k}^{−1}(λ_{3}) π_{k}^{3} & σ_{K|k}^{−1}(λ_{2}) π_{k}^{2} \\ ⋮ & & ... & & ⋮ \\ σ_{K|k}^{2−n}(λ_{n−2}) π_{k}^{n−2} & σ_{K|k}^{2−n}(λ_{n−3}) π_{k}^{n−3} & & σ_{K|k}^{2−n}(λ_{n}) π_{k}^{n} & σ_{K|k}^{2−n}(λ_{n−1}) π_{k}^{n−1} \\ σ_{K|k}^{1−n}(λ_{n−1}) π_{k}^{n−1} & σ_{K|k}^{1−n}(λ_{n−2}) π_{k}^{n−2} & ⋯ & σ_{K|k}^{1−n}(λ_{1}) π_{k} & σ_{K|k}^{1−n}(λ_{n}) π_{k}^{n} \end{matrix} ) .

The elements of Hom^{∘}(A′, A′) ⊗_{ℚ[π_{K}]}ℚ[π_{K}][X]/(X^{n} − π_{K}) have a unique representation as λ_{1}x + λ_{2}x^{2} + ⋯ + λ_{n}x^{n} with λ_{i} ∈ End^{∘}(A′). Under (7), this element corresponds to the first row in the above matrix, i.e. to the row vector

( λ_{n}π_{k}^{n} λ_{n−1}π^{n−1} ⋯ λ_{1}π_{k} ).

Now, every element of ⨁_{i=0}^{n−1} Hom^{∘}(σ_{K|k}^{−i}(A′), A′) has this form with unique λ_{i}. Thus (7) is an isomorphism. □

Remark 4 Since the Frobenius endomorphism has degree a power of p = char(k), we obtain in fact an isomorphism

(End(A′) ⊗_{ℤ[π_{K}]} ℤ[π_{K}][X]/(X^{n} − π_{K})) ⊗ ℤ[1/p] ⟶ End(W) ⊗ ℤ[1/p].

Corollary 5 End^{∘}(W) is commutative if and only if End^{∘}(A′) is commutative.

The isomorphism of Theorem 1 implies that the corresponding centers are isomorphic. Recalling from [10] that Z(End^{∘}(A′)) = ℚ[π_{K}], we thus get:

Corollary 6 We have an isomorphism ℚ[π_{K}][X]/(X^{n}−π_{K}) ≃ Z(End^{∘}(W)).

2.2 Simplicity of the Weil restriction

We are interested in the question whether the Weil restriction W is simple.

In order that W be simple, it is obviously necessary that A′ is simple. Furthermore, it is necessary that A′ is not isogenous to any abelian K-variety which can be defined over any proper intermediate field λ of K|k (i.e. any field λ with k ⊆ λ ⊊ K). (This holds for arbitrary finite separable field extensions K|k.)

For assume that this is the case. Since the scalar restriction of an isogeny is an isogeny, we can assume that A′ itself can be defined over such a λ; A′ = A_{λ} for some λ as above and an abelian λ-variety A. By the defining functorial property of W = Res_{k}^{K}(A′), we have a canonical homomorphism Res_{k}^{λ}(A) ⟶ W which is easily seen to be an immersion. Since the dimension of the immersed abelian variety is strictly smaller, W is not simple.

We thus make the following assumption:

A′ is a simple abelian K-variety which is not isogenous to any abelian K-variety which can be defined over some proper intermediate field λ of K|k.

Lemma 7 Under our assumption on A′, there does not exist a divisor q of n (q ≠ 1) such that π_{K} ∈ ℚ[π_{K}]^{q}.

Proof Assume that such a q exists and let β ∈ ℚ[π_{K}] be such that β^{q} = π_{K}. (In particular ℚ[π_{K}] = ℚ[β].)

Let λ be the subfield of K|k of index q, let V be the Weil restriction of A′ with respect to K|λ. Denoting by χ characteristic polynomials of Frobenius-actions on Tate-modules we have χ_{V} (T) = χ_{A′}(T^{q}), and β is a root of χ_{V}. This follows from the well-known fact that the operation of the absolute Galois group of λ on V (\overline{K}) is induced by the operation of the absolute Galois group of K on A′(\overline{K}); see [5, §1,a)].

It is easy to see that V contains a simple abelian λ-variety A such that the characteristic polynomial of the Frobenius of A has β as a root.

The structure of the endomorphism algebra End^{∘}(A) can be calculated from ℚ[β] as abstract field with generator β; see Subsection 2.3. Inserting β and π_{K} into formula (8), one sees that the central-simple ℚ[π_{K}]-algebras End^{∘}(A) and End^{∘}(A′) have the same local invariants, thus they are isomorphic. Since by formula (9), the dimension of abelian varieties can be calculated from their endomorphism algebras, it follows that dim(A) = dim(A′).

The immersion A ⟶ V = Res_{λ}^{K}(A′) induces by the defining functorial property of the Weil restriction a non-trivial homomorphism A_{K} ⟶ A′. Since the dimensions agree and A′ is simple, this is an isogeny. A contradiction. □

We now make use of the following well-known fact from field theory; see [4, VI, §9, especially Theorem 9.1]:

Lemma 8 Let F be a field, α ∈ F, α ≠ 0 and n ∈ ℕ. Assume that α ∉ F^{q} for all prime divisors q of n. Then either X^{n} − α is irreducible over F or 4|n and α ∈ −4F^{4}.

Together with Corollary 6, this implies:

Proposition 9 Under our assumption on A′,

• either Res_{k}^{K}(A′) has exactly one isotypic component, i.e. all simple abelian subvarieties are isogenous

• or 4|n and π_{K} ∈ −4ℚ[π_{K}]^{4}.

Proof By the previous two lemmata, under our assumption on A′, either X^{n} − π_{K} is irreducible over ℚ[π_{K}] or 4|n and π_{K} ∈ −4ℚ[π_{K}]^{4}. Corollary 6 implies: X^{n} −π_{K} is irreducible over ℚ[π_{K}] if and only if Z(End^{∘}(Res_{k}^{K}(A′))) is a field. This in turn is equivalent to the fact that Res_{k}^{K}(A′) has exactly one isotypic component. □

Remark 10 By Honda’s Theorem (see Proposition 12), it is obviously possible that additionally to our general assumption on A′ the second condition is satisfied. It is interesting to note that there even exist ordinary elliptic curves E′ over fields of the form 𝔽_{p^{4}} (p prime) which are non-isogenous to any elliptic 𝔽_{p^{4}}-curve which can be defined over 𝔽_{p^{2}} and which satisfy

π_{K} ∈ −4ℚ[π_{K}]^{4}. Then Res_{𝔽_{p}}^{𝔽_{p^{4}}}(E′) has more than one isotypic component. Since on the other hand it cannot contain an elliptic curve by our first assumption on E′, Res_{𝔽_{p}}^{𝔽_{p^{4}}}(E′) has exactly two isotypic components both of which are simple.

For example, let p be a prime such that (\frac{−2}{p}) = 1, let K := 𝔽_{p^{4}}, k := 𝔽_{p}.

By assumption, p splits in the field ℚ[\sqrt{−2}]; see [8, Satz 8.5.]. Since this field has class number 1, there is a prime element ν ∈ 𝒪_{ℚ[\sqrt{−2}]} such that (ν)(\overline{ν}) = (p). (Where − denotes conjugation.) Since the norm of an element is always positive, this implies ν\overline{ν} = p. If i ∈ ℕ, then ν^{i} ≠ \overline{ν}^{i}, thus ν^{i} ∉ ℚ.

Let α := −ν^{4}. Then α^{i} ∉ ℚ for all i ∈ ℕ. In particular, ℚ[α] = ℚ[\sqrt{−2}]. Let E′ be a simple abelian K-variety which corresponds to (ℚ[α], α) by Honda’s Theorem (see Proposition 12). By formula (8), all local invariants of End^{∘}(E′) are congruent to 0, thus End^{∘}(E′) ≃ ℚ[α], and E′ is an elliptic K-curve. Since α^{i} ∉ ℚ for all i ∈ ℕ, E′ is ordinary.

The algebraic integer α = −ν^{4} = −4(\frac{ν}{\sqrt{−2}})^{4} lies in −4ℚ[α]^{4}. It remains to check that E′ is not isogenous to any elliptic K-curve which can be defined over 𝔽_{p^{2}}.

Assume this was the case. Then there is a β ∈ End^{∘}(E′) = ℚ[α] = ℚ[\sqrt{−2}] with β^{2} = α = −ν^{4}. This implies i = \sqrt{−1} = \frac{β}{ν^{2}} ∈ ℚ[\sqrt{−2}], a contradiction.

Our aim is now to give conditions under which the Weil restriction of A′ is even simple.

Theorem 2 Let K|k be an extension of finite fields of degree n and A′ a simple abelian K-variety. Assume that A′ is not isogenous to any abelian Kvariety which can be defined over a proper intermediate field of K|k. Assume in addition that one of the following holds:

• End(A′) is commutative and further, if 4|n, then π_{K} ∉ −4ℚ[π_{K}]^{4}.

• The extension degree n is prime.

Then Res_{k}^{K}(A′) is simple.

Proof Assume as in the theorem that A′ is not isogenous to any abelian variety which can be defined over a proper intermediate field of K|k.

We first treat the case that End(A′) is commutative and further, if 4|n, then π_{K} ∉ −4ℚ[π_{K}]^{4}. Under these conditions, End^{∘}(Res_{k}^{K}(A′)) is also commutative (see Corollary 5), and by the above Proposition, Res_{k}^{K}(A′) has exactly one isotypic component. This implies that Res_{k}^{K}(A′) is simple.

We now come to the case that the extension degree n is a prime. Let B ⊆ Res_{k}^{K}(A′) be a simple abelian subvariety. Applying base-change, we get B_{K} ⊆ ∏_{i=0}^{n−1} σ_{K|k}^{−i}(A′). This implies dim(A′) | dim(B). Additionally, the dimensions cannot be equal since otherwise by the defining functorial property of the Weil restriction, we would have an isogeny B_{K} ⟶ A′ which is impossible by assumption. On the other hand, since by Proposition 9 Res_{k}^{K}(A′) has exactly one isotypic component, dim(B) | dim(Res_{k}^{K}(A′)) = n dim(A′). Since n is a prime, this implies dim(B) = dim(Res_{k}^{K}(A′)) thus B = Res_{k}^{K}(A′). □

Remark 11 Let K := 𝔽_{p^{4}}, k := 𝔽_{p} where p is a prime with p ≡ 1 (mod 4). We will now give an elliptic K-curve E′ with non-commutative endomorphism ring such that Res_{k}^{K}(E′) is non-simple even though E′ is not isogenous to any abelian 𝔽_{p^{4}}-variety which can be defined over 𝔽_{p^{2}} and the condition π_{K} ∉ −4ℚ[π_{K}]^{4} is satisfied.

Let E′ be a simple abelian K-variety which corresponds to the integer −p^{2} by Honda’s Theorem; see Proposition 12. By formula (8), the local invariants of End^{∘}(E′) at p and ∞ are congruent to \frac{1}{2}, thus E′ is a supersingular elliptic curve such that all endomorphisms of E′ ; \overline{𝔽}_{p} can be defined over 𝔽_{p^{4}}.

Assume there is an elliptic λ := 𝔽_{p^{2}}-curve E such that E_{K} ∼ E′. Let π_{λ} be its Frobenius endomorphism. Then we have ℚ[π_{λ}] ≃ ℚ[i] (i := \sqrt{−1}), and under this isomorphism, π_{λ} corresponds to ip. Now by assumption, p splits in ℚ[i], and from formula (8), it follows that the local invariants of End^{∘}(E) over p are congruent to \frac{1}{2}, thus by (9), E is 2-dimensional, a contradiction.

Let W be the Weil restriction of E′ with respect to K|k. Then by Corollary 6, the center of End^{∘}(W) is isomorphic to ℚ[X]/(X^{4} + p^{2}) = ℚ[\sqrt[4]{−p^{2}}], and under this isomorphism π_{k} corresponds to \sqrt[4]{−p^{2}}. In this field, p is ramified of degree 2 and splits into 2 prime ideals (because it already splits in the subfield ℚ[i]). Again by formula (8), the endomorphism algebras of the simple components of W are fields, thus isomorphic to ℚ[\sqrt[4]{−p^{2}}]. It follows with (9) that the simple components of W are 2-dimensional, thus W is not simple.

2.3 Appendix to Section 2: Some results by Honda and Tate

For the convenience of the reader, we recall Honda’s Theorem on the classification of simple abelian varieties over finite fields and Tate’s results how to compute the structure of the endomorphism ring of an abelian variety over a finite field; c.f. [3, 10, 11].

Fix a finite field k = 𝔽_{q}, where q = p^{a} with p a prime and a ∈ ℕ. Then, if A is a simple abelian k-variety and π_{k} is its Frobenius endomorphism, for every inclusion φ of ℚ[π_{k}] into \overline{ℚ}, we have |φ(π_{k})| = q^{\frac{1}{2}} .

Now Honda’s Theorem states:

Proposition 12 (Honda) The assignment A ↦ (ℚ[π_{k}], π_{k}) induces a bijection between the set of isogeny classes of simple abelian k-varieties and the set of isomorphism classes of fields ℚ[α] with fixed generator α such that α is an algebraic integer and under all inclusions into \overline{ℚ}, α has absolute value q^{\frac{1}{2}} .

By Honda’s Theorem, for every simple abelian k-variety A, the structure of the endomorphism algebra End^{∘}(A) only depends on ℚ[π_{k}] as abstract field with generator π_{k}. Since End^{0}(A) is central-simple over ℚ[π_{k}], to determine its structure, we only have to give its local invariants at all finite and real valuations.

The formula for this is as follows: Let v be a normalized valuation of ℚ[π_{k}]. Then, if v is finite, the local invariant of End^{∘}(A) at v is given by

inv_{v} ≡ \frac{v(π_{k})}{a}f_{v} (mod 1),  (8)

where f_{v} denotes the absolute residue degree of ℚ[π_{k}] at v. In particular, if v is a finite valuation which does not lie over the valuation of p, the local invariant is congruent to 0.

If v is real, then the local invariant is congruent to \frac{1}{2}.

Let m be the least common denominator of the local invariants. Then the order of End^{∘}(A) in the Brauer group of ℚ[π_{k}] is m, m^{2} = [End^{∘}(A) : ℚ[π_{k}]], and the dimension of A in given by

dim(A) = \frac{1}{2} m [ℚ[π_{k}] : ℚ].  (9)

3 Results for abelian varieties which can be defined over the base-field

Throughout this section, let K|k be a finite Galois extension of degree n with Galois group G, and let A be an abelian k-variety of dimension d. Let W be the Weil restriction of A_{K} with respect to K|k.

We want to determine the structure of the endomorphism ring of W, and the isogeny decomposition of W over k.

3.1 Arithmetic becomes geometric operation

For any k-scheme Z, G operates on A_{K}(Z_{K}) by τ(P) = τP τ^{−1}. These operations define an automorphism of the functor Z ↦ A_{K}(Z_{K}) from the category of k-schemes to the category of abelian groups. We obtain automorphisms of the representing object W = Res_{k}^{K}(A_{K}) which we denote by a_{τ} for τ ∈ G^{opp}. We thus have a group-homomorphism a : G^{opp} ⟶ Aut(W), τ ⟶ a_{τ}, where Aut(W) denotes the group of automorphisms of the abelian k-variety W.

We want to calculate how a_{τ} ⊗_{k} id_{K} operates on W_{K} ≃ A_{K}^{G^{opp}}.

We have τ(u) = τ(p_{id}) = p_{τ^{−1}} : W_{K} ⟶ A_{K} by (2). The homomorphism a_{τ} of the abelian k-variety W is the W -valued point of W which corresponds to τ(u). So by Subsection 1.2, a_{τ} ⊗_{k} id_{K} = (σ^{−1}(τ(u)))_{σ∈G^{opp}} = (σ^{−1}(p_{τ^{−1}}))_{σ∈G^{opp}} = (p_{τ^{−1}σ})_{σ∈G^{opp}}. (The last equation follows from (2).) We have established:

Lemma 13 a_{τ} ⊗_{k} id_{K} : A_{K}^{G^{opp}} ⟶ A_{K}^{G^{opp}} operates on Z-valued points (any Z) by (P_{σ})_{σ∈G^{opp}} ↦ (P_{τ^{−1}σ})_{σ∈G^{opp}}.

3.2 The endomorphism ring as skew group ring

Lemma 14 Let τ ∈ G^{opp}, λ ∈ End(A_{K}). Then a_{τ} ∘Res_{k}^{K}(λ) = Res_{k}^{K}(τ(λ))∘ a_{τ} ∈ End(W).

Proof Easy calculation on Z-valued points. □

To formulate the result about the structure of the endomorphism ring of W, we need a generalization of the concept of a group ring first.

Definition Let Λ be a ring, G a group, t : G ⟶ Aut(Λ) a group-homo-morphism. The application of t(σ) to some λ ∈ Λ will by denoted by σ(λ). Following [9], we define the skew group ring Λ^{t}[G] to be the following ring:^{1} The underlying abelian group is Λ^{G} with the usual “componentwise” addition. As usual, for τ ∈ G, let τ also denote (δ_{σ,τ})_{σ∈G} ∈ Λ^{G}.The multiplication is defined by ∑_{σ∈G} λ_{σ} σ · ∑_{ν∈G} μ_{ν} ν = ∑_{σ,ν∈G} λ_{σ} σ(μ_{ν}) σν.

The ring Λ is naturally immersed in Λ^{t}[G]. For fixed Λ, G and t : G ⟶ Aut(Λ), the ring Λ^{t}[G] has the following universal property:

Lemma 15 Let B be a ring, f : Λ ⟶ B be a ring-homomorphism, and let g : G ⟶ B^{∗} be a group-homomorphism. Assume that for λ ∈ Λ, τ ∈ G, g(τ) f(λ) = f(τ(λ)) g(τ). Then there is a unique ring-homomorphism Λ^{t}[G] ⟶ B with λ ↦ f(λ) and τ ↦ g(τ).

Now let G be the Galois group as above, t : G^{opp} ⟶ Aut(End(A_{K})) the natural operation given by σ ↦ (λ ↦ σ(λ) = σλσ^{−1}). From Lemmata 14 and 15 it follows that ∑_{σ∈G^{opp}} λ_{σ} σ ↦ ∑_{σ∈G^{opp}} Res_{k}^{K}(λ_{σ}) a_{σ} defines a ring-homomorphism

End(A_{K})^{t}[G^{opp}] ⟶ End(W).  (10)

Theorem 3 Let K|k be a finite Galois extension with Galois group G, A an abelian k-variety, W the Weil restriction of A_{K} with respect to K|k, t : G^{opp} ⟶ Aut(End(A_{K})) the natural operation. Then

End(A_{K})^{t}[G^{opp}] ⟶ End(W), ∑_{σ∈G^{opp}} λ_{σ} σ ↦ ∑_{σ∈G^{opp}} Res_{k}^{K}(λ_{σ}) a_{σ}

is an isomorphism.

1This ring is a special case of a crossed product (with respect to some operation); cf.

[9]. In [2], the same ring is called twisted group ring. However, in [9], this word is reserved for the special case of a crossed product with respect to a trivial group operation.

Proof Analogously to the proof of Theorem 1, we make use of the isomorphism Hom(W, W) ≃ Hom(A_{K}^{G^{opp}}, A_{K}) ≃ ⨁_{σ∈G^{opp}} Hom(A_{K}, A_{K}) of the right-hand side.

By (2), the image of some σ ∈ G^{opp} in Hom(A_{K}^{G^{opp}}, A_{K}) is p_{σ^{−1}}, corresponding to the row vector which is zero except at the “σ-th” entry where it is 1.

Thus the image of ∑_{σ∈G^{opp}} λ_{σ}σ (where λ_{σ} ∈ End(A_{K})) is ∑_{σ∈G^{opp}} λ_{σ^{−1}} p_{σ}, corresponding to the row vector (λ_{σ^{−1}})_{σ∈G^{opp}}.

It is thus immediate that we have an isomorphism. □

Corollary 16 The isomorphism in the theorem induces an isomorphism End^{∘}(A_{K})^{t}[G^{opp}] ⟶ End^{∘}(W).

By the Complete Reducibility Theorem (see [6, Proposition 12.1]) we know that the ring End^{∘}(W) is semi-simple. Thus the skew group ring End^{∘}(A_{K})^{t}[G^{opp}] is semi-simple.

It can be proven more generally that every crossed product over a semisimple ring with a finite group in which the group order is invertible is semi-simple; see [9, Theorem 4.1.].

We now want to study the ring-homomorphism

End(A_{K})^{t}[G^{opp}] ⟶^{∼} End(W) ↪
End(W_{K}) ≃ End(A_{K}^{G^{opp}}) ≃ M_{G^{opp}}(End(A_{K})).

(11)

We denote the matrix corresponding to a_{τ} by A_{τ} and the matrix corresponding to Res_{k}^{K}(λ) by J(λ) (for a_{τ} as above and λ ∈ End(A_{K})).

We have already shown in Subsection 1.3 that J(λ) is the diagonal matrix (σ^{−1}(λ)δ_{σ,ν})_{σ,ν∈G^{opp}}.

Let us determine to which matrix A_{τ} ∈ M_{G^{opp}}(End(A_{K})) the endomorphism a_{τ} corresponds. First of all, p_{σ} : W_{K} ≃ A_{K}^{G^{opp}} ⟶ A_{K} corresponds to the row vector (δ_{σ,ν})_{ν∈G^{opp}}. As a_{τ} = (p_{τ^{−1}σ})_{σ∈G^{opp}} (see Lemma 13), we get

A_{τ} = (δ_{τ^{−1}σ,ν})_{σ,ν∈G^{opp}} = (δ_{σ,τν})_{σ,ν∈G^{opp}}.  (12)

Before continuing let us recall the definition of the left regular (matrix) representation.

The left regular (matrix) representation

Let Λ be a ring. If Λ ⟶ Ξ is a homomorphism of rings, we can regard Ξ as Λ-right module, and if we do so, we write End_{Λ}^{r}(Ξ) for the ring of endomorphisms.

Now let Λ ⟶ Ξ be a homomorphism of rings and assume additionally that Ξ is free as Λ-right module on a finite set of generators Σ, i.e. Ξ ≃ Λ^{Σ} as Λ-right modules. Multiplication by elements of Ξ from the left induces a ring-homomorphism

l : Ξ ⟶ End_{Λ}^{r}(Ξ) ≃ End_{Λ}^{r}(Λ^{Σ}),  (13)

the left regular representation.

For a fixed basis Σ, the right-hand side of (13) is canonically isomorphic to the matrix ring M_{Σ}(Λ). The isomorphism is given as follows:

End_{Λ}^{r}(Λ^{Σ}) ⟶ M_{Σ}(Λ), a ↦ (α_{σ,ν})_{σ,ν∈Σ} with α_{σ,ν} ∈ Λ

and a(ν) = ∑_{σ∈Σ} σ α_{σ,ν}. (14)

By composition of (13) with (14), we get the left regular matrix representation (with respect to the basis Σ).

L : Ξ ⟶ M_{Σ}(Λ).

We now apply these concepts in the context of the skew group ring. Let G be a finite group, t : G ⟶ Aut(Λ) be a homomorphism, Λ^{t}[G] the corresponding skew group ring.

We calculate explicitly the left regular representation l : Λ^{t}[G] ⟶ End_{Λ}^{r}(Λ^{t}[G]) and the left regular matrix representation L : Λ^{t}[G] ⟶ M_{G}(Λ) with respect to the basis G.

Let τ ∈ G. Then l(τ) : ν ↦ τν = ∑_{σ∈G} σδ_{σ,τν} and thus

L(τ) = (δ_{σ,τν})_{σ,ν∈G}.

Let λ ∈ Λ. Then l(λ) : ν ↦ λ ν = ν ν^{−1}(λ) and thus

L(λ) = (σ^{−1}(λ) δ_{σ,ν})_{σ,ν∈G}.

We are now going to relate these definitions and calculations with our situation. So let Λ := End(A_{K}), G the Galois group and t : G^{opp} ⟶ End(A_{K}) the natural operation. Let L be the left regular matrix representation of Λ with respect to the basis G^{opp}. Then L(τ) = A_{τ} and L(λ) = J(λ). Thus:

Proposition 17 Homomorphism (11) is the left regular matrix representation of the skew group ring End(A_{K})^{t}[G^{opp}] with respect to the basis G^{opp}.

3.3 The Rosati involution

Let φ : A_{K} ⟶ Â_{K} be a polarization. Then Res_{k}^{K}(φ) : W ⟶ Ŵ is also a polarization; see Subsection 1.5.

We want to calculate how the Rosati involution of W with respect to Res_{k}^{K}(φ) is given under the isomorphism of Corollary 16.

Let us denote the Rosati involution by (…)′.

First of all, the (defining) equation λ′ = φ^{−1}λ̂φ where λ ∈ End^{∘}(A_{K}) implies

Res_{k}^{K}(λ′) = Res_{k}^{K}(φ)^{−1} ∘ Res_{k}^{K}(λ̂) ∘ Res_{k}^{K}(φ) = Res_{k}^{K}(λ)′.

(This holds more generally for any abelian K-variety A′ instead of A_{K}.)

We use the inclusion of End^{∘}(W) into the matrix ring M_{G^{opp}}(End^{∘}(A)) and the fact that Res_{k}^{K}(φ) ⊗_{k} id_{K} is a product polarization to calculate the Rosati involution of a_{τ} with the help of Lemma 3.

Since a_{τ} corresponds to the matrix A_{τ} = (δ_{σ,τν})_{σ,ν∈G^{opp}} (see (12)), a′_{τ} corresponds to the matrix (δ_{ν,τσ})_{σ,ν∈G^{opp}} = (δ_{τ^{−1}ν,σ})_{σ,ν∈G^{opp}} = (δ_{σ,τ^{−1}ν})_{σ,ν∈G^{opp}} = A_{τ^{−1}}. Thus

a′_{τ} = a_{τ^{−1}}.

Since the Rosati involution is an anti-ring-endomorphism, this implies:

Proposition 18 Let K|k be a finite Galois field extension with Galois group G, A an abelian k-variety, W the Weil restriction of A_{K} with respect to K|k. Let φ : A ⟶ Â be a polarization. Let λ ↦ λ′ be the Rosati involution associated to φ. Then under the isomorphism of Corollary 16, the Rosati involution associated to the polarization Res_{k}^{K}(φ) : W ⟶ Ŵ is given by ∑_{σ∈G^{opp}} λ_{σ} σ ↦ ∑_{σ∈G^{opp}} σ^{−1}λ′_{σ} = ∑_{σ∈G^{opp}} σ^{−1}(λ′_{σ}) σ^{−1}.

3.4 Dimensions of components

As in the above proposition, let A be an abelian k-variety, K|k a galois field extension of degree n with galois group G, W the Weil restriction of A_{K} with respect to K|k, and let t : G^{opp} ⟶ End(A_{K}) be the natural operation.

Assume D ⊆ End^{∘}(A_{K}) is a skew field, invariant under the operation t. Let ⨁_{i=1}^{s} Λ_{i} = D^{t}[G^{opp}] be a decomposition of the D^{t}[G^{opp}]-right mod-

ule D^{t}[G^{opp}]. This defines a decomposition 1 = ∑_{i} e_{i} where the e_{i} are orthogonal idempotents, e_{i} ∈ Λ_{i}, such that Λ_{i} = e_{i} D^{t}[G^{opp}]. Conversely, if we are given a decomposition 1 = ∑_{i} e_{i} with orthogonal idempotents e_{i}, then the Λ_{i} := e_{i} D^{t}[G^{opp}] define a direct sum decomposition of the D^{t}[G^{opp}]-right module D^{t}[G^{opp}].

Via the inclusion D^{t}[G^{opp}] ↪ End^{∘}(A_{K})^{t}[G^{opp}] ≃ End^{∘}(W), we can regard the e_{i} to be elements of End^{∘}(W). For each i, let c_{i} ∈ ℕ such that c_{i}e_{i} ∈ End(W).

Now put W_{i} := (c_{i}e_{i})(W). The W_{i} are abelian subvarieties of W and ⨁_{i=1}^{s} W_{i} ∼ W. (Conversely, such an isogeny decomposition where the W_{i} are abelian subvarieties of W determines a decomposition of End^{∘}(W) as right-End^{∘}(W) module.)

Proposition 19 Let D ⊆ End^{∘}(A_{K}) be a skew field, invariant under the operation t on End^{∘}(A_{K}). Let ⨁_{i=1}^{s} Λ_{i} = D^{t}[G^{opp}] be a decomposition of the D^{t}[G^{opp}]-right module D^{t}[G^{opp}]. This corresponds to a decomposition id_{A_{K}} = ∑_{i} e_{i}. Let W_{i} := (c_{i}e_{i})(W) be as above. Then W_{i}_{K} ≈ A_{K}^{n_{i}} (noncanonical isomorphism) where

n_{i} = dim_{D}(Λ_{i}).

Proof Choose a bijection between G^{opp} and the set {1, …, n}. Then A_{K}^{G^{opp}} ≃ A_{K}^{n}.

Let l and L be the left regular (matrix) representations of End^{∘}(A_{K})^{t}[G^{opp}], l_{D} and L_{D} the left regular (matrix) representations of D^{t}[G^{opp}] (both regular matrix representations with respect to the basis G^{opp}). Let ι_{M} : M_{G^{opp}}(D) ⟶ M_{G^{opp}}(End(A_{K})) be the canonical inclusion. Then L = ι_{M} L_{D}.

By construction l_{D}(e_{i}) is the identity on Λ_{i} and zero on all Λ_{j} for j ≠ i. Let n_{i} be the dimension of the D-module Λ_{i}. For each i, choose a basis

(b_{i}^{(j)})_{j=1,...,n_{i}} of the D-module Λ_{i}. Then all n elements b_{i}^{(j)} define a basis of the D-module D^{t}[G^{opp}]. With respect to this basis, the matrix associated to l_{D}(e_{i}) is zero outside a block of size n_{i} where it is the identity matrix.

We now have two matrix representations of l_{D}(e_{i}) with respect to different bases, and via a base change matrix, we can transform one into the other: There exists an invertible matrix B ∈ Gl_{n}(D) such that BL_{D}(e_{i})B^{−1} is zero outside a block of size n_{i} where it is the identity matrix.

Let b ∈ End^{∘}(A_{K}^{G^{opp}}) ≃ End^{∘}(A_{K}^{n}) correspond to ι_{M}(B). By Proposition 17 and our notational conventions, the endomorphism associated to the matrix L(e_{i}) = ι_{M}L_{D}(e_{i}) is e_{i} ⊗_{k} id_{K}. By the above considerations, b(e_{i} ⊗_{k} id_{K})b^{−1} is an endomorphism whose image is isomorphic to A_{K}^{n_{i}}. It follows that the image of c_{i} e_{i} ⊗_{k} id_{K} is also isomorphic to A_{K}^{n_{i}}. □

Remark 20 Let A_{K} be simple, D = End^{∘}(A_{K}). Assume that all e_{i} in the above proposition are central. Then all Λ_{i} as above are rings and we have an isomorphism ∏_{i=1}^{s} Λ_{i} ≃ D^{t}[G^{opp}] ≃ End^{∘}(W) of rings. Furthermore, the (c_{i}e_{i})(W) are generated by isotypic components of W and Λ_{i} ≃ End^{∘}((c_{i}e_{i})(W)). So in particular, the number n_{i} in the above proposition satisfies n_{i} = dim_{D}(End^{∘}(W_{i})).

3.5 The cyclic case

We now apply the above results to the case that G is cyclic of order n.

We identify G with G^{opp} and fix some generator σ ∈ G. Let a = a_{σ} ∈ End(W) be the automorphism corresponding to σ.

Denote the residue class of X in ℚ[X]/(X^{n} − 1) by x. Then we have an inclusion

ℚ[X]/(X^{n} − 1) ⟶ End^{∘}(A_{K})^{t}[G], x ↦ σ.

The polynomial X^{n} − 1 ∈ ℤ[X] splits over ℤ as

X^{n} − 1 = ∏_{d|n} Φ_{d},

where Φ_{d} is the d-th cyclotomic polynomial.

Let Φ′_{d} := (X^{n} − 1)/Φ_{d}. By the Euclidian algorithm, there exist Ψ_{d} ∈ ℚ[X] with ∑_{d|n} Ψ_{d}Φ′_{d} = 1. Let E_{d} := Ψ_{d} Φ′_{d}. Then the E_{d}(x) ∈ ℚ[X]/(X^{n}− 1) are pair-wise orthogonal idempotents. The corresponding decomposition is

ℚ[X]/(X^{n} − 1) ≃ ∏_{d|n} ℚ[X]/Φ_{d} = ∏_{d|n} ℚ(ζ_{d}).

(This is nothing but the Chinese Remainder Theorem in this particular case.)

Let W_{d} := c_{d} E_{d}(a)(W) for suitable c_{d} ∈ ℕ. We then have an isogeny decomposition

W ∼ ∏_{d|n} W_{d},

and by Proposition 19, the W_{d} are abelian varieties with W_{d}_{K} ≈ A_{K}^{φ(d)}.

We also have W_{d} = Φ′_{d}(a)(W). – We only have to show that c_{d}Φ′_{d}(a)(W) ⊆ W_{d}. This follows from Φ′_{d}(x) = (∑_{f|n} Ψ_{f}(x)Φ′_{f}(x))Φ′_{d}(x) = Ψ_{d}(x)Φ′_{d}^{2}(x) = E_{d}(x)Φ′_{d}(x).

It is clear that W_{d} is also the reduced identity component of the kernel of

\begin{matrix}c_{d}(id − E_{d}(a)) = c_{d} ∑_{f|n,f≠d} Ψ_{f}(a)Φ′_{f}(a) = \\ (c_{d} ∑_{f|n, f≠d} Ψ_{f}(a) ∏_{g|n, g≠d,f} Φ_{g}(a)) Φ_{d}(a). \end{matrix}

It is also the reduced identity component of the kernel of Φ_{d}(a). – We only have to show that W_{d} is contained in this kernel. But since W_{d} = Φ′_{d}(a)(W) and Φ′_{d}(x)Φ_{d}(x) = 0, this is obvious.

We now want to study whether the W_{d} are simple or split further. We make the following assumptions.

A_{K} is a simple abelian K-variety whose endomorphisms can be defined over k and whose endomorphism ring is commutative.

Note that if k is finite, all endomorphisms of A_{K} can automatically be defined over k if we assume End(A_{K}) to be commutative.

Also if A is an ordinary elliptic curve over an arbitrary field k and n is odd, then all endomorphisms of A_{K} can be defined over k. This is because under this condition, End(A_{K}) is either ℤ or a quadratic order, thus the only possible non-trivial automorphism of End(A_{K}) has order 2, and consequently the image of the representation Gal(K|k) ⟶ Aut(End(A_{K})) is trivial.

Under the assumptions, we have the isomorphisms

End^{∘}(A)[X]/(X^{n} − 1) ≃ End^{∘}(A_{K})[G] ≃ End^{∘}(W) x  ↦  σ  ↦  a .

Let Φ_{d} split into the product of the non-trivial monic irreducible polynomials Φ_{d}^{(1)}, Φ_{d}^{(2)}, …, Φ_{d}^{(r_{d})} over End^{∘}(A). Let Φ′_{d}^{(i)} := (X^{n} − 1)/Φ_{d}^{(i)}. Since X^{n} − 1 is separable in characteristic zero, the Φ_{d}^{(i)} are pair-wise coprime for varying d and i, and there exist Ψ_{d}^{(i)} with ∑_{d|n} ∑_{1=i}^{r_{d}} Ψ_{d}^{(i)}Φ′_{d}^{(i)} = 1. Let E_{d}^{(i)} := Ψ_{d}^{(i)}Φ′_{d}^{(i)}.

Let W_{d}^{(i)} := c_{d}^{(i)}E_{d}^{(i)}(a)(W) for suitable c_{d}^{(i)} ∈ ℕ. Then again by Proposi-

tion 19, W_{d}^{(i)} is an abelian k-variety with (W_{d}^{(i)})_{K} ≈ A_{K}^{deg(Φ_{d}^{(i)})}. The abelian k-variety W_{d}^{(i)} is simple and its endomorphism algebra is isomorphic to the field End^{∘}(A)[X]/Φ_{d}^{(i)}. The W_{d}^{(i)} are pair-wise non-isogenous (since End^{∘}(W) is commutative), thus they are the isotypic components of W.

As above, one sees that W_{d}^{(i)} = Φ_{d}^{′(i)}(a)(W) and that W_{d}^{(i)} is the reduced identity component of the kernel of Φ_{d}^{(i)}(a).

The component W_{d} is simple if and only if Φ_{d} is irreducible over End^{∘}(A), i.e. if and only if End^{∘}(A)⊗_{ℚ}ℚ(ζ_{d}) is a field. If we fix an inclusion of End^{∘}(A) into \overline{ℚ}, this is the case if and only if End^{∘}(A) ∩ ℚ(ζ_{d}) = ℚ.

In particular, none of the W_{d} splits further if End^{∘}(A) = ℚ as is the case if A is an elliptic curve without complex multiplication (over k).

We proved:

Theorem 4 Let K|k be a cyclic field extension of degree n. Let A be an abelian k-variety.

Let W be the Weil restriction of A_{K} with respect to K|k. For all d|n, W contains canonically an abelian subvariety W_{d} with W_{d}_{K} ≈ A_{K}^{φ(d)} (noncanonically), and W is isogenous to the product of the W_{d}. Here, W_{1} = A itself.

Assume in addition that A_{K} is simple, End^{∘}(A_{K}) is commutative and all endomorphisms of A_{K} can be defined over k. Then the isotypic components of W are all simple, and its endomorphism rings are all commutative. Fix an inclusion of End^{∘}(A_{K}) into \overline{ℚ}. Then for each d, W_{d} is simple if and only if End^{∘}(A) ∩ ℚ(ζ_{d}) = ℚ.

Let E be an ordinary elliptic k-curve with complex multiplication (over k). Fix an inclusion of End^{∘}(E) into \overline{ℚ}. As End^{∘}(E) is a quadratic extension of ℚ, End^{∘}(E) ∩ ℚ(ζ_{d}) ⊋ ℚ if and only if End^{∘}(E) ⊆ ℚ(ζ_{d}). If this is the case then Φ_{d} splits into two polynomials of degree \frac{1}{2}φ(d) over End^{∘}(E).

Corollary 21 Under the assumptions of the theorem, let E be an ordinary elliptic k-curve with End(E) = End(E_{K}) (this condition is automatically satisfied over finite fields or if n is odd).

Then for each d, W_{d} is not simple if and only if E has complex multiplication over k and End^{∘}(E) ⊆ ℚ(ζ_{d}). If this is the case, W_{d} contains two simple non-isogenous abelian subvarieties of dimension \frac{φ(d)}{2}, and W_{d} is isogenous to the product of these abelian subvarieties.

In partular, if n is prime and n ≡ 1 mod 4 or n ≡ 3 mod 4 and \sqrt{−n} ∉ End^{∘}(E), we have an isogeny decomposition W ∼ E × N where N is simple.

Acknowledgements

The results are part of the doctoral thesis of the first author. The thesis was supported by the Friedrich-Naumann-Stiftung, the Deutsche Forschungsgemeinschaft DFG and the Volkswagen Stiftung.

Special cases of some of the present results were obtained in the graduate thesis of the second author.

We would like to thank G. Frey for introducing us the the subject and for constant support during the preparation of our theses.

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Claus Diem: Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Essen, Germany. e-mail: diem@exp-math.uni-essen.de

Niko Naumann: Mathematisches Institut der Universität Münster, Münster, Germany. e-mail: naumannn@uni-muenster.de