Special Values and Transcendence

主要研究人员在算术几何和超越数论中提出了几个项目，这些项目集中于来自特殊函数值的解析信息和算术信息之间的深层联系。在一个项目中，研究人员计划继续研究与Anderson-Drinfeld动机相关的正特征的量，包括周期、对数、Gamma值、Zeta值、多Zeta值和L值。利用差分方程组的伽罗瓦理论，他将寻求关于这些量的代数独立性的结果，既有个别的，也有组合的。在另一个项目中，研究者将考虑将函数域算术不变量与Goss-Hecke L级数的特殊值联系起来的新问题。数论是数学的一个基本分支，它是密码学和编码学等许多应用的基础。这项拟议的研究考虑了涉及解析函数的值的问题，这些值多少显着地传达了关于在其上定义的代数数域或几何对象的基本信息。这样的问题起源于欧拉和高斯的工作，数学家们继续努力解开它们的奥秘。该项目的几个部分自然会导致研究生和本科生研究的问题。

The principal investigator proposes several projects in arithmetic geometry and transcendental number theory that focus on the deep links between the analytic and arithmetic information coming from values of special functions. In one project, the investigator plans to continue the study of quantities related to Anderson-Drinfeld motives in positive characteristic, including periods, logarithms, Gamma values, zeta values, multiple zeta values, and L-values. Using the Galois theory of difference equations, he will pursue results about the algebraic independence of these quantities both individually and in combinations. In another project, the investigator will consider new problems which associate function field arithmetic invariants to special values of Goss-Hecke L-series. Finally, the investigator will pursue problems relating finite field hypergeometric functions to L-series associated to Siegel modular forms.Number theory is one of the fundamental branches of mathematics, and it serves as the basis for many applications, including cryptography and coding theory. The proposed research considers questions involving values of analytic functions that somewhat remarkably convey fundamental information about fields of algebraic numbers or geometric objects defined over them. Such questions have their genesis in work of Euler and Gauss, and mathematicians continue to endeavor to unravel their mysteries. Several parts of the project lead naturally to problems for graduate and undergraduate research.