On finiteness of endomorphism rings of abelian varieties

TL;DR

This paper proves that for fixed-dimension abelian varieties over algebraically closed fields of characteristic p>0, the p-exponents of the co-indices of their endomorphism rings are uniformly bounded, with some applications.

Contribution

It establishes a uniform bound on the p-exponents of co-indices of endomorphism rings for fixed-dimension abelian varieties in characteristic p>0.

Findings

p-exponents of co-indices are bounded for fixed dimension

finiteness result applies to various algebraic contexts

applications to endomorphism ring classification

Abstract

The endomorphism ring End(A) of an abelian variety A is an order in a semi-simple algebra over Q. The co-index of End(A) is the index to a maximal order containing it. We show that for abelian varieties of fixed dimension over any algebraically closed field of characteristic p>0, the p-exponents of the co-indices of their endomorphism rings are bounded. We also give a few applications to this finiteness result.