Raising the Levels of Modular Representations

Abstract

Let ℓ be a prime number, and let F be an algebraic closure of the prime field F _ℓ. Suppose that

$$\rho :Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL(2,\mathbb{F})$$

is an irreducible (continuous) representation. We say that ρ is modular of level N, for an integer N≥ 1, if ρ arises from cusp forms of weight 2 and trivial character on Γ₀(N).