## STNB2024(37th edition)

### On simple reductions of abelian varieties

#### Presenters

Enric Florit

#### Abstract

Let $k$ be a number field and let $A$ be an abelian variety defined over
$k$.
We say $A$ splits if it is isogenous to a product of abelian varieties
of smaller dimension. Otherwise, $A$ is simple. When $A$ is simple, it
may well happen that A splits modulo some prime $\mathfrak p$ of $k$.

A conjecture of Murty and Patankar relates the endomorphism ring
$\operatorname{End}(A)$ to the set of primes of $k$ where $A$ splits.
For example, when $\operatorname{End}(A)$ is noncommutative, it is known
that $A$ splits for a set of primes of density one.

In this talk, we will characterize noncommutative endomorphism algebras
of simple abelian varieties over finite fields. More concretely, we will
use a Theorem of Yu that characterizes the existence of an embedding
$D\hookrightarrow B$ between central simple algebras $D$ and $B$. With
our characterization we are able to prove that, when
$\operatorname{End}(A)$ is noncommutative, A splits modulo all but
finitely many primes $\mathfrak p$ of $k$.

#### Files

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