Arithmetic Applications of the Geometry of Shimura Varieties

志村品种几何学的算术应用

• 批准号: 2200694
• 负责人: Elena Mantovan
• 金额: $ 33.63万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200694&HistoricalAwards=false
• 关键词: Arithmetic; Applications; Geometry; Shimura; Varieties

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.

数论是数学中最古老的分支之一，主要研究整数的性质及其推广，包括素数及其分布问题。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整数解及其几何性质。近年来，志村簇已成为算术几何的主要研究对象，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。这个项目追求新的应用理论的志村品种的一些中心问题的研究算术几何。研究项目包括为本科生和研究生提供培训机会。朗兰兹的对应关系探索伽罗瓦表示和自守形式之间的联系。该项目的研究将开发新的方法来建立同余和构造算术设置中的自守形式的p-adic族，特别是在酉Shimura簇上，并将应用于相关的Galois表示和Serre权结构的研究。算术肖特基问题是指研究交换簇中曲线的雅可比行列式。研究了雅可比矩阵正特征的离散不变量及其随素数变化的行为。理论志村品种自然发生在这种情况下，当人们专注于研究雅可比曲线与额外的自同构。 主要研究者将证明推广埃尔基的里程碑式的结果存在无限多素数的超奇异减少椭圆曲线的雅可比曲线的亏格4和更高。 在新的例子中，模曲线在Elkies的工作中所起的关键作用将由合适的一元志村曲线来发挥，志村曲线被认为是投影线的循环覆盖的特殊家族，在四个点处分支。这个奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。