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

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 the somewhat surprising result 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 $\mathbb{Q}(\sqrt{5})$.