Automorphic Distributions, L-functions, and Cryptography

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This award 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 applications of number theory and discrete groups to cryptography. The first project involves the study of automorphic distributions, and their analytic properties. The PI and Schmid have developed an alternative method for proving the holomorphy of Langlands L-functions, which establishes their holomorphy in new cases. It also has been used to develop Voronoi-style summation formulas, which were later applied to obtain the first subconvexity results for GL(3) L-functions. The project involves extending their core analytic technique of pairing automorphic distributions to more general situations. The second project seeks to develop new algorithms and cryptographic primitives using information from modern number theory, ranging from discrete groups to L-functions.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. The proposed research mainly involves establishing further analytic properties of L-functions, primarily with applications to Langlands' holomorphy conjectures in mind. Such information can be used, as in the second project, to give explicit bounds and parameter estimates for cryptosystems. Work of the PI and Venkatesan on this topic has already found cryptographic applications in numerous Microsoft products. Indeed, modern cryptosystems are often based on hard mathematical problems such as factoring large integers. The proposal seeks to study these problems from the point of view of SAT solvers and coding theory, and to create cryptographic algorithms for other applications.