Hilbert modular forms with prescribed ramification

TL;DR

This paper derives an asymptotic formula for counting Hilbert modular cusp forms with prescribed ramification, revealing that most forms exhibit all possible ramification behaviors, and applies this to determine minimal fields for semi-stability.

Contribution

It provides the first asymptotic count of Hilbert modular forms with specified ramification, generalizing previous results and linking ramification patterns to semi-stability fields.

Findings

Most Hilbert modular forms exhibit all ramification types

Derived an asymptotic formula for counting such forms

Computed minimal fields for semi-stability of modular Jacobians

Abstract

Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the weight of $f$ at $k$, and when $v$ is finite, this means specifying the restriction to inertia of the local Weil-Deligne representation attached to $f$ at $v$. Our formula shows that with essentially finitely many exceptions, the cusp forms of $K$ exhibit every possible sort of ramification behavior, thus generalizing a theorem of Khare and Prasad. From this fact we compute the minimal field over which a modular Jacobian becomes semi-stable.