Arithmetic and Transcendence of Values of Special Functions

这个项目涉及数论和算术几何的研究。这是一个研究领域，通过密码学和编码理论的某些方面应用于网络安全。该研究包括算术几何和超越数论的一些项目，重点是了解特殊解析函数的值如何传达有关代数数和几何对象定义的领域的基本信息。研究人员计划研究数量与安德森-德林费尔德动机和马勒措施的多变量多项式，以具体表现出这种相互作用之间的分析和算术信息。该项目的几个部分自然会给学生的研究带来问题。对于函数域上的特殊值问题，研究者将研究正特征的Anderson-Drinfeld动机的周期和周期，从而发现zeta函数、多重zeta函数、Goss L-函数和Drinfeld多项式的值的新结果。调查的一条线将是应用伽罗瓦理论的弗罗贝纽斯差分方程找到新的结果代数独立的L-值和多个zeta值在任意基环。研究的另一个途径是揭示Drinfeld模及其张量幂的对数代数幂级数恒等式，从而产生Goss L-函数及其扭曲值的新公式，说明它们与Taelman类模的联系。研究人员还将继续研究多变量多项式在复数上的马勒测度，即经典模形式的L函数的特殊值。

This project concerns investigations in the theory of numbers and arithmetic geometry. This is an area of research which has applications to cyber security, through cryptography, and to some aspects of coding theory. The research includes a number of projects in arithmetic geometry and transcendental number theory that focus on understanding how values of special analytic functions convey fundamental information about fields of algebraic numbers and geometric objects defined over them. The investigator plans to study quantities associated to Anderson-Drinfeld motives and to Mahler measures of multivariable polynomials in order to exhibit concretely this interplay between analytic and arithmetic information. Several parts of the project lead naturally to problems for student research. For problems on special values over function fields, the investigator will study periods and logarithms of Anderson-Drinfeld motives in positive characteristic so as to discover new results about values of zeta functions, multiple zeta functions, Goss L-functions, and Drinfeld polylogarithms. One line of inquiry will be to apply the Galois theory of Frobenius difference equations to find new results on algebraic independence of L-values and multiple zeta values over arbitrary base rings. Another path of investigation will be to uncover log-algebraic power series identities for Drinfeld modules and their tensor powers so as to produce new formulas for values of Goss L-functions and their twists that illustrate their connections with Taelman class modules. The investigator will also pursue research on Mahler measures of multivariable polynomials over the complex numbers in terms of special values of L-functions of classical modular forms.