# Weyl bound for trilinear periods via conformal bootstrap

## TL;DR

This paper applies conformal bootstrap techniques to estimate decay rates of modular form coefficients and derives new Weyl bounds for triple product L-functions in the spectral aspect, connecting physics-inspired methods with number theory.

## Contribution

It introduces a novel approach using conformal bootstrap to obtain bounds on triple product L-functions, advancing the understanding of automorphic forms and their associated L-values.

## Key findings

- Established decay rates of modular form coefficients in Laplace eigenbasis.
- Derived new Weyl bounds for triple product L-functions in the spectral aspect.
- Connected conformal bootstrap methods with number theoretic bounds.

## Abstract

Let $f_1,f_2$ be holomorphic modular forms of the same weight for a cocompact lattice $\Gamma < \mathrm{PSL}_2(\mathbf{R})$. We estimate the rate of decay of the coefficients in the expansion of $f_1\overline{f_2}$ in a Laplace eigenbasis. By specializing our main theorem to the case where $\Gamma$ is arithmetic, we obtain new instances of the Weyl bound for triple product $L$-functions in the spectral aspect. Our method builds on the conformal bootstrap in physics.