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 $k$.
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