[en] 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.
Disciplines :
Mathematics
Author, co-author :
Kiming, Ian; University of Copenhagen
Rustom, Nadim; University of Copenhagen
Wiese, Gabor ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
yes
Language :
English
Title :
On certain finiteness questions in the arithmetic of modular forms
Publication date :
2016
Journal title :
Journal of the London Mathematical Society
ISSN :
1469-7750
Publisher :
London Mathematical Society, London, United Kingdom