Endomorphism fields of abelian varieties

2016/06/09 by Guralnick, Robert, Kedlaya, Kiran S. · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1606.02803

Abstract

We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato-Tate group of the abelian variety, which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland. The proof uses Minkowski's reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.

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• Bounds on the Minkowski constants and a function involving φ

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