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 Gal(Q(A[2])/Q) is cyclic of prime order p = 2dim(A) + 1 , we prove that there are only finitely many possibilities for the geometric endomorphism algebra End(A) ⊗ Q . In fact, when dim(A) ∉ {3, 5, 9, 21, 33, 81} , we show End(A) ⊗ Q is a proper subfield of the p-th cyclotomic field. In particular, when g = 2 , End(A) ⊗ Q is isomorphic to either Q or [Formula: see text] .