L-Functions and Rational points

Abstract

This document concerns an application for a BEPE-FAPESP postdoc scholarship at University of California, Irvine, under supervision of professor Daqing Wan from January 2019 till December 2019. As I have written in the research plan of the FAPESP postdoc, process 2017/21259-3, one of the main goals for the postdoc at ICMC-USP, is to expand my research area to the theory of rational points of varieties over finite fields. That is why, I have chosen to work at ICMC and principally with professor Herivelto Borges.Questions about rational points of varieties over finite fields came to me naturally from my doctoral work. In my Ph.D thesis, I have worked with some graphs attached to Hecke operators (over a smooth projective curve defined over a finite field), called "graphs of Hecke operators". Those graphs encodes the action of Hecke operators on automorphic forms and are oriented and weighted. In some situations, the weighted (or multiplicity) of an edge with connect to vertices depends on the number of rational points of that curve. Throughout the first months of my postdoc at ICMC-USP, I am developing together professor Herivelto Borges a project in rational points theory. Namely, we want to improve the well-known bounds for the number of rational points over certain curves defined over a finite field. This project has with main ingredient the work of professor Daqing Wan.Therefore, the main goal in visit professor Daqing Wan, is to use some of his recent work to improve the bound for the number of rational points on certain curves defined over a finite field. Besides that, since professor Daqing Wan is an expert in the theory of L-functions, I intend to learn from his experience the connection between this theory to others, such as the relationship of moment zeta functions and rational points.