A subconvexity bound for Hecke L-functions

Let K be the imaginary quadratic field Q (root -D) with D a prime number congruent to 3 modulo 4, and let psi be a Hecke character on K with conductor (root -D). Let L(s, psi) be the corresponding Hecke L-function. We prove the upper bound L(s, psi) = 0(epsilon)(/S/(13/8) D7/16 + epsilon) which is valid for every epsilon > 0, uniformly in D and s with Rs = 1/2. Note that the exponent in D is strictly less than the exponent 1/2 which can be deduced classically from the functional equation for L(s, psi). It is the first result of that type for such L-functions. (C) 2001 Editions scientifiques et medicales Elsevier SAS.