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 extension generated by its 2-torsion. When $K=Q$ and $\text{Gal}\left(Q\right(A\left[2\right])/Q)$ is cyclic of prime order $p=2dim\left(A\right)+1$ , we prove that there are only finitely many possibilities for the geometric endomorphism algebra $\text{End}\left(A\right)\otimes Q$ . In fact, when $dim\left(A\right)\notin \{3,5,9,21,33,81\}$ , we show $\text{End}\left(A\right)\otimes Q$ is a proper subfield of the p-th cyclotomic field. In particular, when $g=2$ , $\text{End}\left(A\right)\otimes Q$ is isomorphic to either $Q$ or $Q\left(\sqrt{5}\right)$ .