Sato-Tate Distributions of Jacobian Varieties

该项目的总体目标围绕着数学的一些主要目标：识别，可视化和描述模式;确定哪些结构导致这些模式;以及在不同但相关的对象之间建立联系。主要研究者（PI）将在本项目中研究的问题将使用来自广泛数学领域的理论概念和技术，包括数论，群论和算术几何。更具体地说，本项目将重点研究有限域上多项式方程解的数量模式，以及如何通过研究称为Sato-Tate群的相关代数结构来理解这些模式。通过这个项目，PI将通过两项举措为更广泛的社区做出贡献：共同组织一个研究研讨会，支持历史上数学代表性不足的群体的参与者;并与她所在机构的本科生进行研究。多项式方程的解在代数曲线上产生点，Sato-Tate群在研究这些曲线的L-多项式中起着重要作用。推广的Sato-Tate猜想预言曲线的归一化L-多项式的系数的极限分布收敛于与曲线的Jacobian相关联的Sato-Tate群中的迹的分布。这些分布对于亏格1和亏格2曲线是很好理解的，并且在亏格3中已经有了重要的工作。PI将通过进一步发展Sato-Tate群理论和更高亏格曲线及其雅可比变种的分布来扩展文献。在大于3的亏格中，与雅可比簇相关的霍奇群、霍奇环和Mumford-Tate群会发生某些退化，这会影响Sato-Tate群的单位分量和分量群。PI将开发新的技术来确定这些退化情况下的Sato-Tate群。 该奖项反映了NSF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。

The overarching goals of this project are centered around some of the primary goals of mathematics: identifying, visualizing, and describing patterns; determining what structures lead to these patterns; and making connections between distinct but related objects. The questions that the Principal Investigator (PI) will examine in this project will use theoretical concepts and techniques from a wide range of mathematical areas, including number theory, group theory, and arithmetic geometry. More specifically, this project will focus on studying patterns in the number of solutions of polynomial equations over finite fields, and how these patterns can be understood through the study of related algebraic structures called Sato-Tate groups. Through this project, the PI will contribute to the broader community with two initiatives: co-organizing a research workshop that will support participants from groups that are historically underrepresented in mathematics; and conducting research with undergraduates at her home institution. Solutions to the polynomial equations yield points on algebraic curves, and Sato-Tate groups play an important role in the study of L-polynomials of these curves. The generalized Sato-Tate conjecture predicts that the limiting distributions of coefficients of the normalized L-polynomial of a curve converge to the distributions of traces in the Sato-Tate group associated with the Jacobian of the curve. These distributions are well-understood for genus 1 and 2 curves, and there has been significant work in genus 3. The PI will expand on the literature by further developing the theory of Sato-Tate groups and distributions for higher genus curves and their Jacobian varieties. In genus greater than 3, certain degeneracies can occur for the Hodge groups, Hodge rings, and Mumford-Tate groups associated with the Jacobian varieties, and this can affect both the identity component and the component group of the Sato-Tate group. The PI will develop new techniques for determining Sato-Tate groups in these degenerate cases. Both theoretical and computational methods will be utilized in this project.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.