This grant involves two main research projects: a long-term joint project with Wilfried Schmid (Harvard University) on automorphic distributions, and collaborations with Ramarathnam Venkatesan (Microsoft Research) on cryptographic applications of analytic number theory. The first project seeks to use the boundary value distributions of automorphic forms to establish analytic properties of Langlands L-functions. The PI and Schmid have developed an alternative method for proving the holomorphy of Langlands L-functions which bypasses some obstacles existing methods face, and which has resulted in new examples of entire Langlands L-functions (such as the exterior square L-functions on GL(n,Z)GL(n,R)). The project involves extending these results to wider families, and generalizing to number fields and nonarchimedean places. The second project uses results from analytic number theory and representation theory to analyze and create cryptographic schemes, in particular ones related to elliptic curves, isogenies, expander graphs, and modular forms.
L-functions are a central topic in modern number theory, arising from problems as diverse as classical questions about solving polynomial equations with integer solutions, to studying the properties of waves on curved surfaces. Langlands' deep functoriality conjectures assert that these L-functions are all connected to automorphic forms, highly symmetric functions on matrix groups. In particular, they predict the L-functions should be analytic on the complex plane. The proposed research involves establishing more cases of such analytic continuations. The analyticity, once proven, is then known to imply strong results about the original object. Such information can be used, as in the second project, to give explicit bounds and parameter estimates for cryptosystems. In fact, preliminary work of the PI and Venkatesan on the second project has recently been released into security products by the Microsoft Corporation. Modern cryptosystems are very often based on difficult mathematical topics, such as factoring integers and elliptic curves over finite fields. The use of analytic number theory and in particular L-functions is typically essential in gaining both concrete and theoretical understanding of their security.