Explicit small image theorems for residual modular representations

Abstract

Let [Formula: see text] be the residual Galois representation attached to a newform [Formula: see text] and a prime ideal [Formula: see text] in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals [Formula: see text] such that [Formula: see text] is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to [Formula: see text], [Formula: see text] or [Formula: see text]. We also develop explicit criteria to check the reducibility of [Formula: see text], leading to an algorithm that computes the exact set of such [Formula: see text]’s. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz’ [Formula: see text] operator in characteristic zero, and we prove a new Sturm bound theorem.