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.

数论是数学最古老的分支之一，涉及整数性质及其概括的研究，包括有关素数及其分布的问题。算术代数几何是数论的现代子领域，研究多项式方程的积分解及其几何性质。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（如费马大定理及其推广）的重要工具。该项目追求志村簇理论在算术几何中一些核心问题研究中的新应用。研究项目包括为本科生和研究生提供培训机会。朗兰兹的信件探讨了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在算术环境中建立同余和构造自守形式的 p-adic 族，特别是酉志村簇，并将应用于相关伽罗瓦表示和塞尔权猜想的研究。算术肖特基问题是指阿贝尔簇中曲线的雅可比行列式的研究。该研究的重点是研究雅可比行列式的离散不变量的正特征及其随素数变化的行为。当人们专注于研究具有额外自同构的曲线的雅可比行列式时，志村簇理论自然会在这种背景下出现。 首席研究员将证明 Elkies 的里程碑式结果的推广，即椭圆曲线超奇异约简到 4 格及以上曲线的雅克比行列式的存在性。 在新的实例中，模曲线在 Elkies 的工作中发挥的关键作用将由合适的酉 Shimura 曲线发挥，这些曲线被认为是投影线循环覆盖的特殊族，在四个点分支。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。