Weak Subconvexity for Central Values of L-functions

A fundamental problem in number theory is to estimate the values of L-functions at the center of the critical strip. The Langlands program predicts that all L-functions arise from automorphic representations of GL(N) over a number field, and moreover that such L-functions can be decomposed as a product of primitive L-functions arising from irreducible cuspidal representations of GL(n) over Q. The L-functions that we consider will either arise in this manner, or will be the Rankin-Selberg L-function associated to two irreducible cuspidal representations. Note that such Rankin-Selberg L-functions are themselves expected to arise from automorphic representations, but this is not known in general. Given an irreducible cuspidal automorphic representation π (normalized to have unitary central character), we denote the associated L-function by L(s, π), and its analytic conductor (whose definition we shall recall shortly) by C(π). There holds generally a convexity bound of the form L( 1 2 , π) ≪ǫ C(π) 1 4+ǫ (see Molteni [28]). The Riemann hypothesis for L(s, π) implies the Lindelöf hypothesis: L( 1 2 , π) ≪ C(π). In several applications it has emerged that the convexity bound barely fails to be of use, and that any improvement over the convexity bound would have significant consequences. Obtaining such subconvexity bounds has been an active area of research, and estimates of the type L( 1 2 , π) ≪ C(π) 1 4−δ for some δ > 0 have been obtained for several important classes of L-functions. However in general the subconvexity problem remains largely open. For comprehensive accounts on L-functions and the subconvexity problem we refer to Iwaniec and Sarnak [21], and Michel [27]. In this paper we describe a method that leads in many cases to an improvement over the convexity bound for values of L-functions. The improvement is not a saving of a power of the analytic conductor, as desired in formulations of the subconvexity problem. Instead we obtain an estimate of the form L( 12 , π) ≪ C(π) 1 4 /(logC(π))1−ǫ, which we term weak subconvexity. In some applications, it appears that a suitable weak subconvexity bound would suffice in place of genuine subconvexity. In particular, by combining sieve estimates