Attaching `-adic Representations to Elliptic Modular Forms Introduction

In his famous Bourbaki talk [2], Deligne described a recipe for attaching `-adic Galois representations to elliptic modular forms of integral weight at least 2. As a consequence of the method, one reduces the Ramanujan–Petersson conjecture to the validity of Weil’s Riemann Hypothesis for varieties over finite fields. There seems to exist no brief and precise outline of Deligne’s recipe in circulation, and this note is intended to close this gap in the literature. A long and thorough explanation, with complete proofs, may be found in [1]. Fix a integers k ≥ 0 and N ≥ 1. Let f ∈ Sk+2(Γ1(N)) be a Hecke eigenform, with eigenvalues Tpf = apf for p N , and Nebentypus character . Set Kf = Q({ap}), let ` be a rational prime, and choose a prime λ of Kf lying over `. We will construct an continuous homomorphism ρf,λ : Gal(Q/Q) −→ GL2(Kf,λ) that is unramified at all p `N , and for such p satisfies