Formes Modulaires Et Représentations -adiques

(2π)Γ(s)Lτ (s) is invariant under s 7→ 12 − s. The Ramanujan conjecture affirms that the roots of the polynomial Hp are of absolute value p −11/2 (i.e. that |τ(p)| < 2p). These properties, proven or conjectural, are similar to conjectural properties of zeta functions of algebraic varieties over Q. Suggested by this, in a first approximation, it is tempting to interpret the function Lτ as the zeta function of one such variety. For each prime number `, let K` be the maximal extension of Q unramified away from ` and, for p 6= `, let Fp the inverse in the Galois group Gal(K`/Q), of the Frobenius element φp relative to p. This last object is well defined up to conjugation. Translating the preceding in terms of `-adic cohomology, Serre has conjectured the existence, for each `, of a representation of Gal(K`/Q) in a Q`-vector space W` of rank 2 such that, for each p 6= `, one has Hp(X) = det (1− FpX;W`) .