Automorphic forms, L-functions and the relative trace formula

相对迹公式是研究自守形式和朗兰兹纲领的重要工具。Jacquet发明了相对迹公式来研究自守形式的周期积分，特别是，建立了某些情况下的函子性的准则。最近，在某些情况下，工作已经做了明确的L-函数的特殊值的周期的价值。相对迹公式的重要性已得到广泛的承认，而且这个领域还远未穷尽。PI计划将Waldspurger和Jacquet将L值与周期积分联系起来的结果推广到更高级别的群体，扩展她在次凸界上的工作，并将她在平均L值公式上的结果推广到更高级别的L函数，其中已知的要少得多。此外，她还将继续研究有限域上曲线的统计计算，研究随着Artin-Schreier曲线族属的增加，该曲线族的zeta函数的零点分布。PI的提案包括一项计划，指导纽约城市学院的本科生并与他们进行研究。这所大学的学生群体多样化，包括大量传统上代表性不足的群体，包括种族、族裔和社会经济背景。此外，PI将继续她的工作，指导妇女在数学。

The relative trace formula is an important tool in the study of automorphic forms and the Langlands program. Jacquet invented the relative trace formula to study period integrals of automorphic forms, in particular, to establish criteria for certain cases of functoriality in terms of non-vanishing of periods. More recently, work has been done in some cases relating the value of the period explicitly to special values of L-functions. The importance of the relative trace formula is widely recognized and the field is far from exhausted. The PI plans on generalizing to higher rank groups a result of Waldspurger and Jacquet linking L-values with period integrals, extending work of hers on a subconvex bound and generalizing results of hers on an average L-value formula to higher degree L-functions where much less is known. In addition, she will continue her work in computing statistics for curves over finite fields by looking at the distribution of the zeros of the zeta functions of a family of Artin-Schreier curves defined over a finite field as the genus of the family increases.The PI's proposal includes a plan to mentor and conduct research with undergraduate students at The City College of New York. This institution has a diverse student body consisting of large numbers of traditionally underrepresented groups, in terms of race, ethnicity and socio-economic background. In addition the PI will continue her work mentoring women in mathematics.