Functoriality and L-functions

研究者正在进行自守表示理论的研究。 该计划是找到不同的方法来产生自守函数，然后将这些方法联系起来。 找到这种联系的要点是确定自同构L函数的特殊值的自同构型公式（如周期或θ积分）。 具体地说，正则迹公式（以及相对迹公式）的技巧和匡威定理的使用产生了这种函子提升的例子。研究人员将这些技术与θ对应和下降法联系起来。 这些新技术使得确定新提升的非零性成为可能。 上述工作的动机是有一个分析的形式主义，以研究某些自守L函数的特殊值。 事实证明，一些最著名的问题在数论（如费马的最后定理，更普遍的是确定积分解决方案的多项式身份）有作为一个目标的具体分析等L功能如上所述。研究者的方法所提供的信息可能有助于这些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.