# On the rank and the convergence rate towards the Sato-Tate measure

## TL;DR

This paper investigates how quickly the distribution of Frobenius traces of abelian varieties over number fields converges to the Sato-Tate measure, linking convergence rates to arithmetic invariants like rank and Sato-Tate group.

## Contribution

It extends Sarnak's techniques to study convergence rates of Sato-Tate characters for abelian varieties over number fields under standard conjectures, connecting these rates to arithmetic invariants.

## Key findings

- Convergence rate depends on invariants like rank and Sato-Tate group.
- Numerical evidence supports theoretical predictions.
- Methods adapt techniques from elliptic curve analysis to broader abelian varieties.

## Abstract

Let $A$ be an abelian variety defined over a number field and let $G$ denote its Sato-Tate group. Under the assumption of certain standard conjectures on $L$-functions attached to the irreducible representations of $G$, we study the convergence rate of any virtual selfdual character of $G$. We find that this convergence rate is dictated by several arithmetic invariants of $A$, such as its rank or its Sato-Tate group $G$. The results are consonant with some previous experimental observations, and we also provide additional numerical evidence consistent with them. The techniques that we use were introduced by Sarnak, in order to explain the bias in the sign of the Frobenius traces of an elliptic curve without complex multiplication defined over $\mathbb{Q}$. We show that the same methods can be adapted to study the convergence rate of the characters of its Sato-Tate group, and that they can also be employed in the more general case of abelian varieties over number fields. A key tool in our analysis is the existence of limiting distributions for automorphic $L$-functions, which is due to Akbary, Ng, and Shahabi.