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，通过密码学应用于网络安全，并应用于编码理论的某些方面。这项研究包括许多算术几何和超越数论的项目，这些项目侧重于了解特殊解析函数的值如何传达关于代数数域和定义在其上的几何对象的基本信息。研究人员计划研究与Anderson-Drinfeld动机和多变量多项式的Mahler测量相关的量，以便具体地展示分析信息和算术信息之间的这种相互作用。该项目的几个部分自然会给学生的研究带来问题。对于函数域上的特定值问题，作者将研究Anderson-Drinfeld动机的正特征周期和对数，从而发现关于Zeta函数、多重Zeta函数、Goss L函数和Drinfeld多重数的值的新结果。一条途径是应用Frobenius差分方程的伽罗瓦理论，找到关于任意基环上的L值和多重zeta值的代数独立性的新结果。另一条研究途径将是揭示Drinfeld模及其张量幂的对数代数幂等式，从而给出戈斯L函数及其扭曲的值的新公式，以说明它们与Taelman类模的联系。根据经典模形式的L函数的特殊取值，研究复数上多元多项式的马勒测度。

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.