Shimura Varieties with Parahoric and Deeper Level Structure

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The Langlands program is a far-reaching web of conjectures linking seemingly unrelated subjects in mathematics: number theory, representation theory, analysis, and algebraic geometry. More specifically, the Langlands reciprocity conjectures seek to uncover mysterious identities relating functions of arithmetic/number-theoretic nature with functions of analytic/representation-theoretic nature. Such identities reveal hidden symmetries in the objects that give rise to the functions, and each instance of such identities usually has deep consequences. For example, a special case of Langlands' reciprocity conjectures was proved by Wiles and his school, finally resulting in a proof of Fermat's Last Theorem. Fermat's Last Theorem concerned integer solutions of a particular polynomial equation. More general solution sets of polynomials are called algebraic varieties, and among those, Shimura varieties have been central in establishing cases of Langlands' vision. Their study brings together techniques from diverse fields and challenges mathematicians from many different directions. This project centers on the study Shimura varieties as a testing ground of Langlands' reciprocity conjectures. Along the way, the PI will involve his PhD students and Postdocs, and will continue his work on disseminating fundamental science, and on building interactions between different mathematics departments and between mathematicians with diverse fields of expertise.More specifically, the PI will study Shimura varieties from a geometric and representation-theoretic viewpoint, using recent breakthroughs in various domains, in order to express Hasse-Weil zeta functions of very general Shimura varieties in terms of automorphic L-functions. The focus will be on Shimura varieties of abelian type, and the PI will solve various difficulties arising from the presence of singularities on these varieties. These bad reduction issues are easiest to study in the case of parahoric level structure, which will be investigated first, building on prior work of the PI along with recent methods of Kisin-Shin-Zhu in the good reduction (no singularities) cases. The PI will also address deeper level structure situations, bringing in new p-adic geometry techniques due to Scholze and Fargues-Scholze. Of particular importance will be the Fargues-Scholze geometrization of the local Langlands correspondence, and the way this interacts with the PI's ideas on implementing the Langlands-Kottwitz method for deeper level Shimura varieties. This study of bad reduction phenomena will provide steps toward the ultimate goal of realizing links between Galois representations and automorphic representations in the cohomology of Shimura varieties.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.