Arithmetic properties of Fredholm series for p-adic modular forms

TL;DR

This paper investigates the arithmetic properties of Fredholm series related to p-adic modular forms, proving non-vanishing modulo p at tame level 1 and exploring conjectures about slopes near the boundary of p-adic weight space.

Contribution

It establishes non-vanishing of Fredholm series coefficients modulo p at tame level 1 and examines their behavior at higher levels, connecting to recent modular interpretations.

Findings

Coefficients of Fredholm series never vanish modulo p at tame level 1.

At higher levels, infinitely many coefficients are non-zero modulo p.

Supports conjectures on slopes of overconvergent p-adic modular forms.

Abstract

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.