Explicit Small Image Theorems for Residual Modular Representations

TL;DR

This paper establishes explicit bounds and criteria for residual Galois representations associated with modular forms, enabling the precise identification of exceptional primes through an implemented algorithm.

Contribution

It provides explicit bounds, criteria, and an algorithm for detecting exceptional residual Galois representations, including new theoretical results and computational tools.

Findings

Explicit bounds for exceptional primes are derived.

An algorithm for computing all such primes is developed and implemented.

New theoretical results include lifts of Katz' θ operator and a Sturm bound theorem.

Abstract

Let $\rho$ f,$\lambda$ be the residual Galois representation attached to a newform f and a prime ideal $\lambda$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals $\lambda$ such that $\rho$ f,$\lambda$ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to A4, S4 or A5. We also develop explicit criteria to check the reducibility of $\rho$ f,$\lambda$ , leading to an algorithm that compute the exact set of such $\lambda$. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz' $\theta$ operator in character zero, and we prove a new Sturm bound theorem.