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

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 $\dim (A)\notin \{3,5,9,21,33,81\}$, 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 $\mathbb{Q}(\sqrt{5})$.

17 pages. v4: Final version. To appear in Research in Number Theory. v5: corrected Theorem 3.5