Arithmetic and Transcendence of Values of Special Functions

主要研究者在算术几何和超越数论中提出了几个项目，重点研究来自特殊函数值的解析信息和算术信息之间的深层联系。研究者计划继续研究与Anderson-Drinfeld动机相关的正特征量，特别是周期、对数和特殊的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无关性结果。这些研究的另一个方面将是寻找和发展在德林菲尔德模块及其张量幂上的对数代数幂级数恒等式。对这些恒等式的专门化将产生Dirichlet字符、Hecke字符和Drinfeld模块的Goss l系列的特殊值的显式公式，从而为超越和代数无关问题提供输入。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变异点的计数以及与经典和西格尔模形式相关的l系列的问题。数论是数学的一个基本分支，它是许多应用的基础，包括密码学和编码理论。提出的研究考虑了涉及解析函数值的问题，这些值在某种程度上显著地传达了关于代数数域或定义在它们之上的几何对象的基本信息。这样的问题在欧拉和高斯的著作中有其起源，数学家们一直在努力解开它们的奥秘。这个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.