Automorphic forms, L-functions and the relative trace formula

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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.