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 extension generated by its 2-torsion. When K= Q K = Q and Gal (Q (A2) / Q) Gal (Q (A 2) / Q) is cyclic of prime order p = 2 (A) +1 p = 2 dim (A) + 1, we prove that there are only finitely many possibilities for the geometric endomorphism algebra End (A) Q End (A) ⊗ Q. In fact, when (A) \3, 5, 9, 21, 33, 81\ dim (A) ∉ 3, 5, 9, 21, 33, 81, we show End (A) Q End (A) ⊗ Q is a proper subfield of the p -th cyclotomic field. In particular, when g=2 g = 2, End (A) Q End (A) ⊗ Q is isomorphic to either Q Q or Q (5) Q (5).
Pip Goodman (Mon,) studied this question.