Santiago Arango-Piñeros : $q$-Frobenius distributions of abelian varieties

Posted by Zhehao Li , part of the Graduate Number Theory Seminar.

At

April 5, 2023, 4 p.m.

In

Zoom

Abstract

Fix a $g$-dimensional abelian variety $A$ defined over a finite field $\mathbf{F}_q$. For every integer $r \geq 1$, consider the extension of scalars $A_{(r)} = A\times_{\mathbf{F}_q} \mathbf{F}_{q^r}$. We study the distribution of the normalized trace of the Frobenius endomorphism of $A_{(r)}$ in the compact interval $[-2g,2g]$, as $r$ varies. We show that these distributions are controlled by a certain compact abelian Lie group and classify the possible groups when $g \leq 3$.

Zoom link: https://uic.zoom.us/j/82357749325?pwd=L1JoRm9rUjZJbDNZeUJvdDljWGdhUT09