Families of Automorphic Forms with Prescribed Local Behavior

该研究项目的目标是发展一个定量理论的家庭与规定的地方行为。一般来说，这个想法是，人们可以研究一个数学对象：一个簇，一个表示，或一个L函数，通过变形它在家庭。族是一组共享某些公共局部特征的全局对象。数学家预言，族具有等分布特性，即族的局部成分在规定的空间内以均匀的方式变化。构建这些属性的证明是我们处理这些对象的数学能力的里程碑。此外，这个项目将为研究生提供研究培训的机会。家庭是至关重要的，即使一个人是一个先验感兴趣的单一自守形式。亚瑟在1988年提出，如果一个自守表示有一个局部分量属于超尖点L-包，那么它是调和的。在函数域上，PI建议与Sawin一起建立单式几何超尖点（mgs）包的猜想，即那些由字符的紧归纳产生并在非分歧基变化后保持超尖点的包。该方法依赖于结合曲线上G-丛的模空间的l-adic几何和迹公式。该奖项反映了NSF的法定使命，并已被认为是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的支持。

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.