On mod p modular representations which are defined over Fp

On Mod P Modular Representations Which Are Defined over F P

In this paper, we will use techniques of Conrey, Farmer and Wallace to find spaces of modular forms Sk(Γ0(N)) where all of the eigenspaces have Hecke eigenvalues defined over Fp, and give a heuristic that these are all such spaces.

7 M ay 2 00 9 On mod p representations which are defined over F p : II

Elliptic mod l Galois representations which are not minimally elliptic

In a recent preprint (see [C]), F. Calegari has shown that for l = 2, 3, 5 and 7 there exist 2-dimensional surjective representations ρ of Gal(Q̄/Q) with values in Fl coming from the l-torsion points of an elliptic curve defined over Q, but not minimally, i.e., so that any elliptic curve giving rise to ρ has prime-to-l conductor greater than the (prime-to-l) conductor of ρ. In this brief note, w...

MOD p REPRESENTATIONS ON ELLIPTIC CURVES

Modular Galois representations ρ : Gal(Q/Q) → GL2(Fp) with cyclotomic determinant arise from elliptic curves for small p. We show that ρ does not necessarily arise from an elliptic curve whose conductor is as small as possible outside p. For p = 3 this disproves a conjecture of J. Lario and A. Rio [8].

ON MODULAR MOD l GALOIS REPRESENTATIONS WITH EXCEPTIONAL

We give a parametrization of the possible Serre invariants (N, k, ν) of modular mod ℓ Galois representations of the exceptional types A 4 , S 4 , A 5 , in terms of local data attached to the fields cut out by the associated projective representations. We show how this result combined with certain global considerations leads to an effective procedure that will determine for a given eigenform f a...