A classical question in number theory is to find solutions in rational numbers or integers to systems of polynomial equations. In the twentieth century, mathematicians realized that this question can be reformulated using geometric objects called algebraic cycles. This realization gave rise to a vast and beautiful conjectural framework, which now includes some of the most important unresolved conjectures in mathematics. On the other hand, the discovery of the law of quadratic reciprocity (due to Gauss) and its generalizations lead ultimately to the formulation of the Langlands program, which is a separate web of conjectures relating the symmetries of numbers to analysis and group theory. The questions to be studied in this project lie at the interface of these two different webs of conjectures, and thus involve objects of enormous arithmetic richness. The specific goal of the project is to use the study of certain highly symmetric functions to reveal information about the geometry and arithmetic of polynomial equations. While the final goal is to reveal sophisticated information about polynomials, many of the objects to be studied have immediate practical applications. For example, elliptic curves, which are cubic equations in two variables, play a prominent role in this research and also an important role in contemporary applications such as cryptography and digital signatures, which have extensive use in commerce. One of the broader impacts of the project will be the development of a course on the mathematics of cryptocurrencies such as bitcoin, popularizing mathematics through an exciting application of broad current popular interest. The project will also provide research training activities for graduate students
In technical terms, the main thrust of the research is to use the fine structure of automorphic representations, including the theory of local and global Arthur packets, to study problems on algebraic cycles. Specific problems to be studied include: (i) constructing Hodge cycles that represent instances of Langlands functoriality, especially for unitary groups; (ii) Oda's conjecture on the factorization of Hodge structures of Hilbert modular forms; (iii) relations between Abel-Jacobi images of cohomologically trivial cycles and p-adic L-functions; (iv) integral period relations for quaternionic modular forms, and (v) applications of the theory of non-tempered A-packets to generalizations of Kudla-Millson theory on locally symmetric spaces. The investigator will continue to mentor graduate student research on topics related to the themes in the project.
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.
