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

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 $$K= \mathbb {Q}$$ K = Q and $$\textrm{Gal}(\mathbb {Q}(A[2])/\mathbb {Q})$$ Gal ( Q ( A [ 2 ] ) / Q ) is cyclic of prime order $$p = 2 \dim (A) +1$$ p = 2 dim ( A ) + 1 , we prove that there are only finitely many possibilities for the geometric endomorphism algebra $$\textrm{End}(A) \otimes \mathbb {Q}$$ End ( A ) ⊗ Q . In fact, when $$\dim (A) \not \in \{3,5,9,21,33,81\}$$ dim ( A ) ∉ { 3 , 5 , 9 , 21 , 33 , 81 } , we show $$\textrm{End}(A) \otimes \mathbb {Q}$$ End ( A ) ⊗ Q is a proper subfield of the p -th cyclotomic field. In particular, when $$g=2$$ g = 2 , $$\textrm{End}(A) \otimes \mathbb {Q}$$ End ( A ) ⊗ Q is isomorphic to either $$\mathbb {Q}$$ Q or $$\mathbb {Q}(\sqrt{5})$$ Q ( 5 ) .