THE SUBCONVEXITY PROBLEM FOR L-FUNCTIONS

Abstract

Estimating the size of automorphic L-functions on the critical line is a central problem in analytic number theory. An easy consequence of the standard analytic properties of theL-function is the convexity bound, whereas the generalised Riemann Hypothesis predicts a much sharper bound. Breaking the convexity barrier is a hard problem. The moment method has been used to surpass convexity in the case of Lfunctions of degree one and two. In this talk I will discuss a different method, which has been quite successful to settle certain longstanding open problems in the case of degree three. At the 1994 International Congress at Zürich, J. B. Friedlander [1995] briefly described the essence of the amplified moment method which he was developing in a series of joint works with Duke and Iwaniec, with the aim of obtaining non-trivial bounds for Lfunctions. Since then the amplification technique has proved to be very effective in a number of scenarios involvingGL(2)L-functions (see J. Friedlander and Iwaniec [1992], Duke, J. B. Friedlander, and Iwaniec [1993, 1994, 1995, 2001, 2002], Kowalski, Michel, and VanderKam [2002], Michel [2004], Harcos and Michel [2006], and Blomer and Harcos [2008]). But there are major hurdles in extending the method far beyond. In the last decade the automorphic period approach has been developed in great detail and generality (over number fields), by Michel, Venkatesh and others (see Bernstein and Reznikov [2010], Michel and Venkatesh [2010], Wu [2014]). This puts the moment method in a proper perspective and gives a satisfactory explanation to the ‘mysterious identities between families of L-functions’ that already occurs in the study of the moments of the Rankin-Selberg L-functions Harcos and Michel [2006], Michel [2004]. This has been the topic of Michel’s address at the 2006 International Congress at Madrid Michel and Venkatesh [2006]. Here I will briefly describe a new approach to tackle subconvexity, which has not only settled some of the longstanding open problems in the field, but has also matched in strength the existing benchmarks. As there are several excellent accounts MSC2010: primary 11F66; secondary 11M41.