On certain finiteness questions in the arithmetic of modular forms
arXiv:1408.3249 · doi:10.1112/jlms/jdw045
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 Γ_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.
25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and changes
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