On certain finiteness questions in the arithmetic of modular forms

TL;DR

This paper explores finiteness properties of reductions of p-adic Galois representations from modular forms, linking them to conjectures about p-adic coefficient fields and providing partial proofs and numerical evidence.

Contribution

It formulates a new finiteness conjecture for reductions of modular eigenforms modulo prime powers and proves a weaker version with supporting numerical data.

Findings

Proposes a finiteness conjecture for reductions modulo p^m of modular forms.

Establishes a weak version of the conjecture.

Provides numerical evidence supporting the conjecture.

Abstract

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.