Mathematical Sciences: Arithmetic of Automorphic Forms on GSp(4)

这项研究是在所谓的朗兰兹纲领的一般领域，这个主题位于现代分析中数论和表征理论之间的桥梁上。更具体地说，可以将l函数附加到有理数阿戴尔环上二乘二矩阵群的倒自同构表示上，也可以将l函数附加到有理数代数闭包的伽罗瓦群的不可约表示上。在这个项目中要研究的主要问题是这两个l函数何时重合。已经设计了一种新的方法来回答这个问题，并将继续采用这种方法。包含大量数论信息的两种不同类型的函数将被研究。当它们能被证明是一致的时候，人们就能从更容易理解的那一个中获得关于更模糊的那一个的许多重要的算术信息。这些公式的生成是本研究的重点。

This research is in the general area of the so-called Langlands Program, a subject that lies on the bridge between Number Theory and Representation Theory in Modern Analysis. More specifically one attaches an L-function to a cuspidal automorphic representation of the group of two by two matrices over the rational adele ring, and one also attaches an L-function to an irreducible representation of the galois group of the algebraic closure of the rational numbers. The main question to be investigated on this project is when these two L-function coincide. A new method has been devised to answer this question and this method will be pursued. Two different types of functions that contain a great deal of number theoretic information will be investigated. When they can be shown to coincide one can obtain much important arithmetic information about the more obscure one from the better understood one. Producing these formulas is the focus of this research.