Arithmetic Applications of the Geometry of Shimura Varieties

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Number theory is one of the oldest branches of mathematics, and is concerned with the study of properties of the integers and their generalization, including questions about prime numbers and their distribution. Arithmetic Algebraic Geometry is a modern subfield of Number Theory which deals with the study of integral solutions to polynomial equations, and their geometric properties. In recent years, Shimura varieties have become a main object of study in arithmetic geometry, and a crucial tool in the solution of many outstanding problems in Number Theory, such as Fermat's Last Theorem and its generalizations. This project pursues new applications of the theory of Shimura varieties to the study of some central problems in arithmetic geometry. The research projects include training opportunities for both undergraduate and graduate students.The Langlands' correspondences explore the connections between Galois representations and automorphic forms. The research in this project will develop new methods to establish congruences and construct p-adic families of automorphic forms in the arithmetic setting, specifically on unitary Shimura varieties, and will have applications to the study of the associated Galois representations, and Serre weight conjectures. The arithmetic Schottky problem refers to the study of Jacobians of curves among abelian varieties. The research focuses on the study of discrete invariants of Jacobians in positive characteristics and of their behavior as the prime varies. The theory of Shimura varieties naturally occurs in this context when one focuses on the study of Jacobians of curves with extra automorphisms. The principal investigator will prove generalizations of Elkies' landmark result on the existence of infinitely many primes of supersingular reduction for elliptic curves to Jacobians of curves of genus 4 and higher. In the new instances, the crucial role that the modular curve plays in Elkies' work will be played by suitable unitary Shimura curves which are known to arise as special families of cyclic covers of the projective line, branched at four points.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.