[Paper Review] On the Structure of the Weil Restriction of Abelian Varieties

Claus Diem, Niko Naumann|ArXiv.org|Apr 18, 2005

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper provides a complete description of the endomorphism rings and isogeny decompositions of Weil restrictions of abelian varieties over finite Galois extensions of fields. It shows that for a cyclic extension $ K|k $ of degree $ n $, the Weil restriction $ W $ of an abelian variety $ A_K $ decomposes into isotypic components $ W_d $, each corresponding to a divisor $ d $ of $ n $, with $ W_d $ simple if and only if $ \mathbb{Q} \cap \operatorname{End}^\circ(A)\otimes\mathbb{Q}(\zeta_d) = \mathbb{Q} $, generalizing the behavior of elliptic curves with complex multiplication.

ABSTRACT

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

Motivation & Objective

• To determine the structure of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions.
• To study the isogeny decomposition of Weil restrictions, particularly in the case of finite fields and cyclic extensions.
• To provide explicit criteria for when the Weil restriction is simple or decomposes into simple factors.
• To generalize known results on elliptic curves with complex multiplication to higher-dimensional abelian varieties.

Proposed method

• Use of skew group rings to describe the endomorphism ring of the Weil restriction $ W $ as a crossed product over $ \operatorname{End}(A_K) $ with the Galois group $ G^\text{opp} $.
• Application of representation theory and idempotent decomposition via cyclotomic polynomials $ \Phi_d $ over $ \operatorname{End}^\circ(A) $.
• Construction of subvarieties $ W_d^{(i)} $ via projection operators $ \Psi_d^{(i)} \Phi_d^{(i)'} $, where $ \Phi_d^{(i)} $ are irreducible factors of $ \Phi_d $ over $ \operatorname{End}^\circ(A) $.
• Use of the canonical isomorphism $ \sigma_{K|k}: \sigma^{-1}(A_K) \to A_K $ to define Frobenius and Galois actions on the Weil restriction.
• Leveraging the fact that $ \operatorname{End}^\circ(W) \simeq \operatorname{End}^\circ(A_K)[G] $ under the assumption that all endomorphisms of $ A_K $ are defined over $ k $.
• Reduction to the case of cyclotomic polynomials and number field intersections to determine simplicity of components.

Experimental results

Research questions

• RQ1 Under what conditions is the Weil restriction of an abelian variety over a finite Galois extension simple over the base field?
• RQ2 How does the endomorphism ring of the Weil restriction relate to the Galois group and the endomorphism ring of the original variety?
• RQ3 What is the isogeny decomposition of the Weil restriction when the base extension is cyclic?
• RQ4 When does a component $ W_d $ of the Weil restriction remain simple, and when does it split into non-isogenous simple factors?
• RQ5 How does complex multiplication on an elliptic curve affect the isogeny decomposition of its Weil restriction over a cyclic extension?

Key findings

• The endomorphism algebra of the Weil restriction $ W $ of $ A_K $ over $ K|k $ is isomorphic to $ \operatorname{End}^\circ(A_K)[G] $, the skew group ring over $ \operatorname{End}^\circ(A_K) $ with the Galois group $ G^\text{opp} $, provided all endomorphisms of $ A_K $ are defined over $ k $.
• For a cyclic extension $ K|k $ of degree $ n $, the Weil restriction $ W $ decomposes into isogeny components $ W_d $ indexed by divisors $ d \mid n $, with $ W_d $ having $ (W_d)_K \approx A_K^{\varphi(d)} $.
• Each component $ W_d $ is simple if and only if $ \operatorname{End}^\circ(A) \cap \mathbb{Q}(\zeta_d) = \mathbb{Q} $, i.e., the intersection of the endomorphism field with the $ d $-th cyclotomic field is trivial.
• If $ A $ is an ordinary elliptic curve with complex multiplication over $ k $, then $ W_d $ is not simple if and only if $ \operatorname{End}^\circ(A) \subseteq \mathbb{Q}(\zeta_d) $, in which case $ W_d $ splits into two non-isogenous simple factors of dimension $ \varphi(d)/2 $.
• In particular, if $ n $ is prime and $ n \equiv 1 \pmod{4} $ or $ n \equiv 3 \pmod{4} $ with $ \sqrt{-n} \notin \operatorname{End}^\circ(E) $, then $ W \sim E \times N $ with $ N $ simple.
• The isotypic components $ W_d^{(i)} $ are constructed as images of idempotent operators $ E_d^{(i)} = \Psi_d^{(i)} \Phi_d^{(i)'} $, and they are simple abelian $ k $-varieties with endomorphism algebra isomorphic to $ \operatorname{End}^\circ(A)[X]/\Phi_d^{(i)} $.