On certain finiteness questions in the arithmetic of modular forms

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 of normalized eigenforms on Γ1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.