Tian Wang : Distribution problems associated to abelian varieties

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

March 31, 2023, noon

In

612 SEO

Abstract

Given an abelian variety $A$ defined over $\mathbb{Q}$, we look at its reduction $A_p$ at an arbitrary prime $p$ and study the distribution of $A_p$ with specific properties. For example, the Lang-Trotter Conjecture about an abelian variety $A$ of dimension $1$ and its generalizations to an abelian variety $A$ of higher dimension concern the distribution of $A_p$ having a prescribed Frobenius trace, while the Murty-Patankar Conjecture about an absolutely simple abelian variety $A$ concerns the distribution of $A_p$ having split reduction. In this talk, I will discuss new upper bounds for the counting functions related to the aforementioned conjectures.