Special Functions and Transcedence

研究人员在算术几何和超越数论中提出了几个探索解析函数特殊值的算术性质的问题。在一个项目中，研究人员计划研究安德森-德恩菲尔德动机产生的正特征数的超越性和代数独立性。这些超越问题与Galois组的Frobenius差分方程组密切相关，特别是研究者提出追求超越理论和Galois理论之间的相互作用来研究函数域的多重Zeta值、Drinfeld模的周期和对数以及高维形式。作者建议研究模形式的傅立叶系数及其与有限域超几何函数和Calabi-Yau流形的关系。他还计划发现椭圆曲线扩张群的算术性质及其与伽罗瓦表示的联系。作为现代数学的基本分支之一，数论是许多应用的基础，包括密码学和编码理论。这项拟议的研究考虑了涉及数论的算术和分析方面的相互作用的问题，这些问题起源于欧拉和高斯的经典著作，今天继续对数论和许多其他数学分支中的问题的基本理解具有重要影响。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 investigator proposes several problems in arithmeticgeometry and transcendental number theory that explore the arithmeticnature of special values of analytic functions. In one project, theinvestigator plans to study transcendence and algebraic independence ofnumbers in positive characteristic arising from Anderson-Drinfeld motives. These transcendence questions are closely connected to Galois groups of Frobenius difference equations, and in particular the investigatorproposes to pursue the interaction between transcendence theory and Galoistheory to study function field multiple zeta values, periods and logarithms of Drinfeld modules, and higher dimensional versions. The investigator proposes to study Fourier coefficients of modular forms and their relationships with finite field hypergeometric functions and Calabi-Yau manifolds. He plans also to discover arithmetic properties of the extension groups of elliptic curves and their connections with Galois representations.One of the fundamental branches of modern mathematics, number theoryserves as the basis for many applications, including cryptography andcoding theory. The proposed research considers questions involving theinterplay between arithmetic and analytic aspects of number theory, which have their beginnings in classical work of Euler and Gauss and which continue to have important implications today to the essential understanding of problems both in number theory and in many other branches of mathematics. Several parts of the project lead naturally to problems for graduate and undergraduate research.