On modular Galois representations modulo prime powers

TL;DR

This paper explores three levels of modularity for Galois representations mod p^m, establishing their relationships, and demonstrating how stronger notions imply weaker ones, with implications for eigenforms and level reduction.

Contribution

It introduces and compares three notions of modularity for Galois representations mod p^m, and proves a level-lowering result for strongly modular representations.

Findings

Strongly modular representations can be reduced to dc-weakly modular ones at lower levels.

All three notions of modularity coincide when m=1.

A Galois representation can be attached to any dc-weak eigenform under certain conditions.

Abstract

We study modular Galois representations mod $p^m$. We show that there are three progressively weaker notions of modularity for a Galois representation mod $p^m$: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level $M$. Using results of Hida we display a `stripping-of-powers of $p$ away from the level' type of result: A mod $p^m$ strongly modular representation of some level $Np^r$ is always dc-weakly modular of level $N$ (here, $N$ is a natural number not divisible by $p$). We also study eigenforms mod $p^m$ corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod $p^m$ to any `dc-weak' eigenform, and hence to any eigenform mod $p^m$ in any of the three senses. We show that the three notions of modularity coincide when $m=1$ (as well as in other, particular cases), but not in general.