Arithmetic Applications of the Geometry of Shimura Varieties

数论是数学最古老的分支之一，主要研究整数的性质及其推广，包括关于素数及其分布的问题。算术代数几何是数论的一个现代分支，主要研究多项式方程的积分解及其几何性质。志村变分近年来已成为算术几何研究的主要对象，也是解决数论中许多突出问题的重要工具，如费马大定理及其推广。本课题致力于志村变分理论在算术几何中心问题研究中的新应用。研究项目包括本科生和研究生的培训机会。朗兰兹夫妇的书信探讨了伽罗瓦表征和自同构形式之间的联系。本项目的研究将开发新的方法来建立同余和构造自同构形式的p进族的算术设置，特别是在酉Shimura变种，并将有应用于研究相关的伽罗瓦表示，和Serre权猜想。算术肖特基问题是研究阿贝尔变量间曲线的雅可比矩阵问题。主要研究雅可比矩阵的离散不变量的正特征及其随素数变化的行为。当人们关注具有额外自同构曲线的雅可比矩阵的研究时，志村变分理论自然而然地出现在这种背景下。主要研究者将证明Elkies关于椭圆曲线超奇异化简的无穷多素数存在的里程碑式结果推广到4属及以上曲线的雅可比矩阵。在新的实例中，模曲线在Elkies的工作中所起的关键作用将由合适的统一志村曲线来发挥，这些曲线被称为射影线的特殊循环覆盖族，在四个点上分支。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.