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arXiv · 1408.3249

On certain finiteness questions in the arithmetic of modular forms

Ian Kiming·Nadim Rustom·Gabor Wiese
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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.

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Ian Kiming, Nadim Rustom, Gabor Wiese. 2014-08-14. On certain finiteness questions in the arithmetic of modular forms. https://doi.org/10.1112/jlms%2Fjdw045

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