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

TL;DR

This paper investigates the endomorphism algebras of abelian varieties with specific properties over number fields, focusing on cases where the 2-torsion field is large and cyclic.

Contribution

The paper provides new criteria for endomorphisms to be defined over a specific field and identifies finitely many possibilities for the endomorphism algebra in certain cases.

Findings

When the Galois group is cyclic of prime order p = 2 dim(A) + 1, there are finitely many possibilities for the geometric endomorphism algebra.

For dimensions not in {3,5,9,21,33,81}, the endomorphism algebra is a proper subfield of the p-th cyclotomic field.

For g=2, the endomorphism algebra is either Q or Q(sqrt(5)).

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 \documentclass[12pt]{minimal}
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\begin{document}$$K= \mathbb {Q}$$\end{document}K=Q and \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{Gal}(\mathbb {Q}(A[2])/\mathbb {Q})$$\end{document}Gal(Q(A[2])/Q) is cyclic of prime order \documentclass[12pt]{minimal}
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\begin{document}$$p = 2 \dim (A) +1$$\end{document}p=2dim(A)+1, we prove that there are only finitely many possibilities for the geometric endomorphism algebra \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{End}(A) \otimes \mathbb {Q}$$\end{document}End(A)⊗Q. In fact, when \documentclass[12pt]{minimal}
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\begin{document}$$\dim (A) \not \in \{3,5,9,21,33,81\}$$\end{document}dim(A)∉{3,5,9,21,33,81}, we show \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{End}(A) \otimes \mathbb {Q}$$\end{document}End(A)⊗Q is a proper subfield of the p-th cyclotomic field. In particular, when \documentclass[12pt]{minimal}
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\begin{document}$$g=2$$\end{document}g=2, \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{End}(A) \otimes \mathbb {Q}$$\end{document}End(A)⊗Q is isomorphic to either \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Q}$$\end{document}Q or \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Q}(\sqrt{5})$$\end{document}Q(5).