Explicit Small Image Theorems for Residual Modular Representations
Let $ρ$ f,$λ$ be the residual Galois representation attached to a newform f and a prime ideal $λ$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals $λ$ such that $ρ$ f,$λ$ 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 $ρ$ f,$λ$ , leading to an algorithm that compute the exact set of such $λ$. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz' $θ$ operator in character zero, and we prove a new Sturm bound theorem.
math.NT↗