On Mod P Modular Representations Which Are Defined over F P

Weights of Galois Representations Associated to Hilbert Modular Forms

Let F be a totally real field, p ≥ 3 a rational prime unramified in F , and p a place of F over p. Let ρ : Gal(F/F ) → GL2(Fp) be a two-dimensional mod p Galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at p is irreducible. We specify a set of weights, determined by the restriction of ρ to inertia at p, which contains all the ...

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

Deformation of Galois Representations

ρ : GQ,S → GL2(Zp) such that the reduction of ρ mod p is equivalent to ρ̄. Such representations are called deformations of ρ̄ (in what follows, an over-bar means reduction mod p). This question is closely connected with that of congruence between modular forms. Indeed, let f ∈ Sk(Γ0(Np ),Zp) be a cuspidal eigenform (here k ≥ 2, r ≥ 1). Work of Shimura, Deligne, shows that one can associate to f a...

Elliptic Curves with 3-adic Galois Representation Surjective Mod 3 but Not Mod 9

Let E be an elliptic curve over Q, and ρl : Gal(Q/Q)→GL2(Zl) its l-adic Galois representation. Serre observed that for l ≥ 5 there is no proper closed subgroup of SL2(Zl) that maps surjectively onto SL2(Z/lZ), and concluded that if ρl is surjective mod l then it is surjective onto GL2(Zl). We show that this no longer holds for l = 3 by describing a modular curve X9 of genus 0 parametrizing elli...

Modular Calabi–yau Threefolds of Level Eight

If X̃ is a Calabi–Yau threefold defined over Q, and p is a good prime (i.e., a prime such that the reduction of X̃ mod p is nonsingular) then the map Frob∗p : H i ét(X̃,Ql) −→ H i ét(X̃,Ql) on l–adic cohomology induced by the geometric Frobenius morphism gives rise to l–adic Galois representations ρl,i : Gal(Q/Q) −→ GLbi(Ql). If a Calabi–Yau threefold X̃ is rigid (i.e., h(X̃) = 0 or equivalently b(X̃)...