Special Values of L Functions and Functoriality

1.数学家Stephen Rallis的工作提出了两种确定从经典群到Gl(n)的泛函性的方法。这是通过应用逆定理和一个好的L函数的有效理论来完成的。第二种方法是利用相对迹线公式。这些观点是基于早期的自同构血统理论。他以下降的形式主义方法开发了经典群的cap模块构造方法。此外，前面提到的L函数理论也允许将贝塞尔周期的不消失与经典群上张量L函数的不消失联系起来。对于普通大众，Stephen Rallis的研究系统地使用了对称的基本原理来研究数论中的基本问题。令人难以置信的基本问题，如著名的费马最后猜想，就需要这样的想法。斯蒂芬·拉利斯（Stephen Rallis）发展了各种类型的对称（以数学的方式），构成了这门学科。重点是寻找新的方法来证明Gl(n)和其他经典群在函数方程对称中心的自对偶自同构L函数的非负性。他还提出了新的时期条件来分析这些特殊的价值。这对于与L函数在特定点不消失有关的各种算术、解析和几何思想是重要的。这项工作训练研究生的数学研究能力。

1.For Mathematicians Stephen Rallis's work presents two methods to determine functoriality from a classical group to Gl(n). This is done by applying the Converse Theorem coupled with an effective theory of good L functions. The second method is by use of the relative trace formula. The ideas are based on an earlier theory of automorphic descent. He develops with the descent formalism ways to constructCap modules for classical groups. Also the aforementioned theory of L functions allows a procedure to relate nonvanishing of Bessel periods to the nonvanishingof tensor L functions on classical groups.2. For the general public Stephen Rallis's research uses systematically the basic principles of symmetry to investigate fundamental problems in number theory. Incredibly basic problems such as the celebrated Fermat Last Conjecture require such ideas. Stephen Rallis develops various types of symmetry (in a mathematical way) that make up this subject. The main emphasis is to find new ways to show the nonnegativity of self dual automorphic L functions for Gl(n) and other classical groups at the center of symmetry of the functional equation. Also he develops new period conditions to analyze these special values.This is important for the various arithmetic, analytic and geometric ideas tied to nonvanishing of L functions at specific points. This work trains graduate students in mathematical research.