On certain finiteness questions in the arithmetic of modular forms

TY - JOUR

T1 - On certain finiteness questions in the arithmetic of modular forms

AU - Kiming, Ian

AU - Rustom, Nadim

AU - Wiese, Gabor

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

AB - 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.

U2 - 10.1112/jlms/jdw045

DO - 10.1112/jlms/jdw045

M3 - Journal article

SN - 0024-6107

VL - 94

SP - 479

EP - 502

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 2

ER -