Mathematical Sciences: Automorphic Forms and Representation Theory

本研究以几何方法探讨自同构形式与表示理论。主要问题是求出Shimura变Hasse-Weil zeta函数的解析延拓和泛函方程。目的是用自同构l函数来表示这些函数。然后必须得到这些自同构l函数的变换规律。本研究将研究多项式方程组以所有素数为模的解个数的生成函数。这些函数的性质将提供大量有关丢番图方程的解的信息。数论的主要组成部分之一是丢番图方程的研究，即整数方程组的解。

This research deals with the subject of automorphic forms and representation theory using geometric methods. The main problem is to obtain the analytic continuation and functional equation of the Hasse-Weil zeta function of Shimura varieties. The object is to express these zeta functions in terms of automorphic L-functions. Then one must obtain the transformation laws for these automorphic L-functions. This research will study the generating functions for the number of solutions of systems of polynomial equations modulo all the primes. The properties of these functions will give an immense amount of information concerning the solution to the corresponding set of Diophantine Equations. Number Theory has as one of its chief components the study of Diophantine Equations, that is the solution of sets of equations in integers.