Arithmetic and Transcendence of Values of Special Functions

主要研究者提出了算术几何和超越数论的几个项目，重点是来自特殊函数值的分析和算术信息之间的深层联系。 研究人员计划继续研究与安德森-德林费尔德动机有关的正特征量，特别是周期、周期、特殊zeta和L值。 一个重点将是进一步研究差分方程的伽罗瓦理论，以证明这些量的新的代数独立性结果。这些调查的另一个方面将是寻找和发展对数代数幂级数身份Drinfeld模块和他们的张量权力。 这些恒等式的特殊化将为高斯L-级数的狄利克雷特征、赫克特征和德林费尔德模的特殊值产生明确的公式，从而为超越和代数独立性问题提供输入。在另一个项目中，研究者将研究有限域超几何函数与有限域上代数簇上的计数点以及与经典和Siegel模形式相关的L-级数的问题。数论是数学的基本分支之一，它是许多应用的基础，包括密码学和编码理论。 拟议的研究考虑的问题涉及的解析函数的值，有点显着传达基本信息领域的代数数或几何对象定义在他们身上。 这些问题起源于欧拉和高斯的工作，数学家们继续奋进解开它们的奥秘。 该项目的几个部分自然会给研究生和本科生的研究带来问题。

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. The investigator plans to continue the study of quantities related to Anderson-Drinfeld motives in positive characteristic, especially periods, logarithms, and special zeta and L-values. One focus will be to investigate further the Galois theory of difference equations to prove new algebraic independence results for these quantities. Another aspect of these investigations will be to search for and develop log-algebraic power series identities on Drinfeld modules and their tensor powers. Specializations of these identities will then produce explicit formulas for special values of Goss L-series for Dirichlet characters, Hecke characters, and Drinfeld modules, thus providing input for transcendence and algebraic independence problems. In another project, the investigator will pursue problems relating finite field hypergeometric functions to counting points on algebraic varieties over finite fields and L-series associated to classical and 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.