Functoriality and L-functions

这位研究人员正在执行一项关于自同构表示理论的程序。程序是找到不同的方法来产生自同构函数，然后将这些方法联系起来。寻找这种联系的目的是为了确定自同构L函数的特定值的自同构型公式(如周期或西塔积分)。具体地说，正则迹公式(以及相对迹公式)的技巧和逆定理的使用产生了这种函数式提升的例子。研究人员将这些技术与theta对应和下降方法联系起来。这样的新技术使得根据特定值来确定新升降物的非零性成为可能。上述工作的动机是用一种解析形式来研究某些自同构L函数的特殊值。事实证明，数论中一些最著名的问题(如费马的S大定理，以及更一般的多项式恒等式的整数解的确定)都以上述L函数的具体分析为目标。研究人员的方法所提供的信息可能有助于对这些L函数的定性研究。

The investigator is carrying out a program in the theory of automorphic representations. The program is to find different methods to produce automorphic funtoriality and then relate such methods. The point of finding such connections is to determine automorphic-type formulae( such as periods or theta integrals) for special values of automorphic L functions. Specifically the techniques of the regular trace formula (as well as the relative trace formula) and the use of the converse theorem produce examples of such functorial lifting. The investigator relates these techniques to theta correspondence and the descent method. Such new techniques make it possible to determine the nonvanishing of the new liftings in terms of special values. The motivation of the above work is to have an analytic formalism to study the special values of certain automorphic L functions. It turns out that some of the most famous problems in number theory (such as Fermat' s last theorem and more generally the determination of integral solutions to polynomial identities) have as a goal the concrete analysis of such L functions as above. The information given by the investigator's approach may be useful in the qualitative study of these L functions.