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 K = Q and Gal(Q(A[2])/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 ) .