# Identifying central endomorphisms of an abelian variety via Frobenius endomorphisms

## TL;DR

Under the assumption of the Mumford-Tate conjecture, the paper demonstrates how to determine the center of an abelian variety's endomorphism ring using Frobenius endomorphisms and provides an algorithm for its computation.

## Contribution

The paper establishes a method to recover the center of the endomorphism ring from Frobenius data assuming the Mumford-Tate conjecture and introduces a practical algorithm for this purpose.

## Key findings

- The center can be recovered from Frobenius endomorphisms under the Mumford-Tate conjecture.
- An explicit algorithm for computing the center of the endomorphism ring is provided.
- The approach links Frobenius data to the algebraic structure of abelian varieties.

## Abstract

Assuming the Mumford-Tate conjecture, we show that the center of the endomorphism ring of an abelian variety defined over a number field can be recovered from an appropriate intersection of the fields obtained from its Frobenius endomorphisms. We then apply this result to exhibit a practical algorithm to compute this center.