Slopes of overconvergent Hilbert modular forms

TL;DR

This paper explicitly describes the $U_p$ operator on overconvergent Hilbert modular forms, computes slopes for specific weights, and proposes a conjecture on the structure of these slopes based on computational evidence.

Contribution

It provides an explicit matrix description of the $U_p$ operator and introduces a conjecture on the structure of slopes near the boundary of weight space.

Findings

Slopes near the boundary are not unions of arithmetic progressions.

A simple recipe can generate slopes, leading to a new conjecture.

A lower bound on the Newton polygon of $U_p$ is established.

Abstract

We give an explicit description of the matrix associated to the $U_p$ operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the centre and near the boundary of weight space for certain real quadratic fields. \added[id=h]{Near the boundary of weight space we see that the slopes do not appear to be given by finite unions of arithmetic progressions but instead can be produced by a simple recipe from which we make a conjecture on the structure of slopes. We also prove a lower bound on the Newton polygon of the $U_p$.