L-Values, Special Cycles, and Euler Systems

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Number theory, especially the study of properties of integers and rational numbers and the integer or rational solutions to equations, is one of the oldest branches of mathematics. It is also a meeting ground for many other branches of mathematics -- analysis, geometry, representation theory (to name a few) -- and progress on the fundamental problems in number theory continues to reveal surprising connections between these branches as well as new applications to other areas. This project is focused on one of these fundamental problems and aims to establish new instances of these connections.The fundamental problem at the heart of this project is to determine the arithmetic nature of the special values of L-functions of an algebraic variety or motive and to relate these values to the orders of algebraic quantities associated to the variety or motive (such as a Chow group, a class group, or a Selmer group). An important and motivating instance is the celebrated Birch and Swinnerton-Dyer Conjecture for elliptic curves, vastly generalized by the Bloch-Kato Conjectures. This project will extend progress toward this fundamental problem through further development of connections between: L-functions, Galois representations, automorphic forms, representation theory, p-adic variation of L-values and modular forms, special cycles (especially on Shimura varieties), and special classes in Galois cohomology groups. This progress will unfold in two primary directions: (A) the proof of new instances of the Bloch-Kato Conjectures or their consequences (such as new cases of the p-part of the Birch and Swinnerton-Dyer formula), and (B) the development of new tools and new methods for proving results towards these conjectures (such as new applications of representation theory to the existence Euler systems).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.