Applications of the relative trace formula in higher rank

该提案的目标是将自守形式空间上定义的周期积分与L函数的特殊值联系起来。具体而言，共同PI期望通过将周期积分与二次基变化L-函数的中心值相关来将Waldspurger的结果推广到更高的秩。在这项工作中使用的主要工具是由Jacquet发起的相对迹公式。co-PI还计划探索使用相对迹公式研究L-函数族，以了解如何使用相对迹公式来解决次凸性问题。L-函数提供了自守形式和数论世界之间的联系。L-函数的特殊值经常编码重要的算术信息;例如Birch和Swinnerton-Dyer猜想断言椭圆曲线的L-函数决定了关于椭圆曲线结构的重要信息。椭圆曲线已经成为许多研究的焦点，从怀尔斯证明费马大定理到密码学。

The goal of the proposal is to relate period integrals defined on spaces of automorphic forms to special values of L-functions. Specifically the co-PI expects to generalize results of Waldspurger to higher rank by relating period integrals to central values of quadratic base change L-functions. The main tool to be used in this work is the relative trace formula as initiated by Jacquet. The co-PI also plans to explore the use of the relative trace formula in the study of families of L-functions with a view towards understanding how the relative trace formula can be used to attack the subconvexity problem.L-functions provide a connection between the world of automorphic forms and number theory. Special values of L-functions frequently encode important arithmetic information; for example the Birch and Swinnerton-Dyer conjecture asserts that the L-function of an elliptic curve determines important information about the structure of the elliptic curve. Elliptic curves have become a focal point of much research, from Wiles' proof of Fermat's Last Theorem to cryptography.