Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field

TL;DR

This paper investigates the endomorphism algebras of abelian varieties over number fields with large cyclic 2-torsion fields, establishing criteria for endomorphisms to be defined over specific fields and classifying possible endomorphism algebras in certain cases.

Contribution

It provides new criteria for endomorphisms to be defined over the 2-torsion field and classifies endomorphism algebras for abelian varieties with cyclic 2-torsion Galois groups over .

Findings

Finite possibilities for the geometric endomorphism algebra when the Galois group is cyclic of prime order.

Endomorphism algebra is a subfield of the p-th cyclotomic field for most dimensions.

Explicit classification of endomorphism algebras for abelian surfaces.

Abstract

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be defined over $K(A[2])$, the field generated by its 2-torsion. When $K= \mathbb{Q}$ and $\mathrm{Gal}(\mathbb{Q}(A[2])/\mathbb{Q})$ is cyclic of prime order $p = 2 \dim(A) +1$, we prove that there are only finitely many possibilities for the geometric endomorphism algebra $\mathrm{End}(A) \otimes \mathbb{Q}$.In fact, when $\dim (A) \not \in \{3,5,9,21,33,81\}$, we show $\mathrm{End}(A) \otimes \mathbb{Q}$ is a proper subfield of the $p$-th cyclotomic field. In particular, when $g=2$, $\mathrm{End}(A) \otimes \mathbb{Q}$ is isomorphic to either $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{5})$.