Algebraic Cycles and Motivic Cohomology in the Context of the Langlands Program

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This research project lies at the interface of algebraic geometry and number theory. Algebraic geometry is the study of the geometry of shapes cut out by systems of polynomial equations, while in number theory one is often interested in finding explicit solutions to such systems in rational numbers or integers. The problem of finding such solutions is a foundational question in mathematics, having been studied for more than two thousand years, and is a central theme of this proposal. This project will investigate higher dimensional versions of this question in specific contexts arising from modular and automorphic functions. Modular and automorphic functions encode deep arithmetic information and are increasingly playing important roles in other fields, including theoretical physics. This project aims to broaden and deepen knowledge in this fundamental area of mathematics.More specifically, the award involves work on five distinct projects on algebraic cycles in the context of the Langlands program: (i) integral period relations for Hilbert modular forms and their analogs on quaternionic Shimura varieties; (ii) the Bloch-Beilinson conjecture for Rankin-Selberg L-functions and in particular constructing cycles corresponding to the vanishing of the central value; (iii) the construction of absolute Hodge classes corresponding to cases of Langlands functoriality; (iv) the study of the injectivity of the Abel-Jacobi map for zero cycles on surfaces over number fields; (v) the relation between motivic cohomology and the cohomology of automorphic forms. A theme in many of the projects is the construction of explicit algebraic cycles or more generally elements in motivic cohomology, whose existence is often predicted by deep general conjectures on algebraic cycles.