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Families of Automorphic Forms with Prescribed Local Behavior

具有规定局部行为的自守形式族

基本信息

批准号：
2001071
负责人：
Nicolas Templier
金额：
$ 35万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001071&HistoricalAwards=false
关键词：
Families Automorphic Forms Prescribed Local

项目摘要

The goal of the research project is to develop a quantitative theory of families with prescribed local behavior. Generally speaking, the idea is that one can study a mathematical object: a variety, a representation, or an L-function, by deforming it in families. A family is a set of global objects that share some common local features. Mathematicians predict that families have equidistribution properties in the sense that their local components should vary in a uniform way within their prescribed space. Building proofs of such properties is a milestone in our mathematical ability to work with these objects. Additionally this project will provide research training opportunities for graduate students.Families are crucial even if one is a priori interested in a single automorphic form. Arthur conjectured in 1988 that if an automorphic representation has a local component that belongs to a supercuspidal L-packet, then it is tempered. Over function fields, the PI proposes to establish with Sawin this conjecture for monomial geometric supercuspidal (mgs) packets, that is for those packets that arise from compact-induction of characters and remain supercuspidal after unramified base change. The method relies on combining the l-adic geometry of the moduli space of G-bundles on a curve and on trace formulas.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.

该研究项目的目标是发展具有规定的局部行为的家庭的定量理论。一般来说，这个想法是，人们可以研究一个数学对象：一个变量，一个表示，或者一个l函数，通过在家庭中变形它。族是一组具有一些共同局部特征的全局对象。数学家预测，族具有等分布性质，即它们的局部成分应该在它们规定的空间内以统一的方式变化。建立这些性质的证明是我们处理这些对象的数学能力的一个里程碑。此外，该项目将为研究生提供研究培训机会。即使一个人先验地对单一自同构形式感兴趣，家庭也是至关重要的。Arthur在1988年推测，如果一个自同构表示有一个属于超尖l包的局部分量，那么它是调质的。在函数域上，PI提出用Sawin建立单项式几何超尖（mgs）包的这个猜想，即那些由字符的紧归纳产生的包，在非分支碱基变化后仍保持超尖。该方法依赖于将曲线上g束模空间的l进几何与迹公式相结合。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。