Explicit Small Image Theorems for Residual Modular Representations
Abstract
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.
Domains
Number Theory [math.NT]
Origin : Files produced by the author(s)
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