The PI will study various problems in the arithmetic theory of automorphic forms suggested by conjectures on algebraic cycles, notably the Tate conjecture and the Bloch-Beilinson conjecture. One of the problems involves proving integral period relations for arithmetic automorphic forms on quaternionic Shimura varieties. These relations are known up to algebraic factors, due to previous work of Michael Harris. The PI proposes to prove much more precise relations that identify more or less exactly the missing algebraic factors. Such relations would have applications to the theory of special values of L-functions. In addition, the methods used to study this problem are expected to yield new constructions of algebraic cycles. Another project involves studying the relations between cycles and p-adic L-functions, especially for unitary groups. This develops and generalizes a theme studied in the PI's previous work with Bertolini and Darmon.
The general area of this proposal is algebraic number theory. More specifically, it deals with the study of algebraic cycles over number fields which may be thought of as a higher dimensional generalization of the solutions in rational numbers or integers to a given polynomial equation. The study of integer solutions to polynomial equations (also called Diophantine equations) is a problem that has interested people for the last two thousand years. It is such a basic problem in mathematics that finding new insights into it is likely to have many applications, not just to other parts of mathematics but also practical in nature. Some of the key objects that will be studied, namely elliptic curves, have many practical applications to coding theory and cryptography. The projects in the proposal will lead to not just a better theoretical understanding of such objects, but also develop new computational tools to study them.
