On Ordinary Forms and Ordinary Galois Representations

Let p ≥ 5 be a prime. A well-known conjecture of Serre (see [18]) asserts that any two dimensional irreducible representation

ρ : Gal(Q/Q) → GL₂ (F_p),

which is odd (in the sense that det(c) = −1 , where c is complex conjugation), arises from reduction modulo p of the p-adic representation attached to a newform by Deligne (see [2]). In such a situation, we say that ρ is a modular mod p Galois representation.