# Central limit theorems for elliptic curves and modular forms with smooth weight functions

## TL;DR

This paper proves a Central Limit Theorem for smoothed Sato-Tate distributions, enabling results for smaller families of modular forms and elliptic curves under the Riemann Hypothesis, improving previous limitations.

## Contribution

It introduces a smoothed approach to the Sato-Tate conjecture, allowing CLT proofs for smaller families and under the Riemann Hypothesis, advancing the understanding of distributional properties.

## Key findings

- CLT established for smaller families of modular forms.
- CLT proved for elliptic curves under Riemann Hypothesis.
- Overcomes previous limitations with a smoothed method.

## Abstract

The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a Central Limit Theorem for much smaller families of modular forms, and we succeed in proving a theorem of this type for families of elliptic curves under the Riemann Hypothesis for $L$-functions associated to Hecke eigenforms for the full modular group.