Number Theory Seminar

(Joint with Kannan Soundararajan.) We describe a new approach to bounding central values of L-functions which builds on work of Heath-Brown. The method yields an unconditional bound of the form $|L(1/2,f)|\ll C^{1/4}/(\log C)^{\delta}$ for the $L$-function of any $GL(m)$ cusp form $f$, where $C$ is the analytic conductor of $f$ and $\delta$ is a constant; a similar also holds for Rankin-Selberg $L$-functions. The proof relies on a careful study of the distribution of zeros of $L(s,f)$ near $s=1$ with input from diophantine analysis, sieve methods, and Rankin-Selberg theory.