Traces on Hecke algebras and families of p-adic modular forms

TL;DR

This paper proves that finite slope modular forms can be organized into p-adic families indexed by weight, confirming predictions of the Mazur-Gouvea Conjecture through trace formula comparisons.

Contribution

It provides a new proof of the Mazur-Gouvea Conjecture for finite slope forms using trace formulas, differing from Coleman's rigid analytic geometry approach.

Findings

Finite slope modular forms form p-adic families indexed by weight

Dimension of slope subspace is independent of weight

Supports predictions of the Mazur-Gouvea Conjecture

Abstract

In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic congruences between the corresponding modular forms. We also show that the dimension of the slope subspace of the space of modular forms does not depend on the weight. Both statements are predicted by the Mazur-Gouvea Conjecture, which has been proven by Coleman using methods from rigid analytic geometry. In contrast our proof is based on a comparison of (topological) trace formulas.