Explicit large image theorems for modular forms

Abstract

Let k and N be positive integers with k ≥ 2 even. In this paper we give general explicit upper-bounds in terms of k and N from which all the residual representations ρ f,λ attached to non-CM newforms of weight k and level Γ 0 (N ) with λ of residue characteristic greater than these bounds are "as large as possible". The results split into different cases according to the possible types for the residual images and each of them is illustrated on some numerical examples.

Non-optimal levels of a reducible mod $\ell $ modular representation

“…A newform f of level pr = 38 (as above) is E [38,2]. Let a n be the eigenvalue of T n for E [38,2]. Then a q ≡ q + 1 (mod 25) when q = 23,41,97,101,109,113,149,151,193,199,239,241,251,257,277,347,359,431, and 479 for primes q < 500.…”

“…Since only (2, 151), (2, 241), (2, 251), and (2, 431) are admissible for s = 2, a triple (2, q, 19) is admissible for s = 2 if q = 23, 41, 97, 101, 109, 113, 149, 193, 199, 239, 257, 277, 347, 359, or 479 for q < 500. A newform f of level pr = 26 (as above) is E [26,2]. Let d n be the eigenvalue of T n for E [26,2].…”

“…In most cases, the necessary conditions above are also sufficient for the admissibility. In particular, Theorem 1.3 precisely characterizes admissible t-tuples for s when 1 ≤ s ≤ t ≤ 4, except in the cases (s, t) ∈ {(2, 2), (2,3), (4,4)}. In such exceptional cases, an extension of our methods allows us to obtain sufficient conditions, which are more complicated to state.…”

Modular forms of arbitrary even weight with no exceptional primes

“…The first argument of this sort, for the k = 2 case, goes back to Ribet (see the proof of [12, Proposition 2.2]). For higher weights, see [1,Section 3.3], which we follow closely.…”

“…Nevertheless, it can be very hard to explicitly identify the exceptional primes for any given modular form. Recent work of Billerey-Dieulefait gives explicit but complicated bounds on the exceptional primes for a modular form of weight k ≥ 2 and trivial nebentype [1].…”