# A uniform Weyl bound for L-functions of Hilbert modular forms

## TL;DR

This paper proves a uniform Weyl-type subconvexity bound for L-functions of Hilbert modular forms with specific level conditions, using advanced techniques like Motohashi's formula and hypergeometric sums over finite fields.

## Contribution

It introduces a new uniform subconvexity bound for Hilbert modular forms' L-functions under particular level and local representation conditions.

## Key findings

- Established a Weyl-type subconvexity bound for Hilbert modular forms' L-functions.
- Utilized a distributional Motohashi's formula over number fields.
- Applied Katz's work on hypergeometric sums in the proof.

## Abstract

We establish a Weyl-type subconvexity of $L(\tfrac{1}{2},f)$ for spherical Hilbert newforms $f$ with level ideal $\mathfrak{N}^2$, in which $\mathfrak{N}$ is required to be cube-free, and at any prime ideal $\mathfrak{p}$ with $\mathfrak{p}^2 \mid \mathfrak{N}$ the local representation generated by $f$ is not supercuspidal. The proof exploits a distributional version of Motohashi's formula over number fields developed by the first author, as well as Katz's work on hypergeometric sums over finite fields in the language of $\ell$-adic cohomology.