Mathematics 690STE - ZetaFunctions/AlgebraVarieties

Given a collection of polynomial equations defined over Q, we can associate to it a family of generating functions, called zeta functions. They are built using topological construction based on local information (specifically solutions of these equations modulo prime powers). But there are deep and important conjectures predicting that these functions encode global information --- the Q-solutions of these equations. Much of the subject of modern Arithmetic Geometry is about understanding this interplay between local and global objects. The goal of this course is to survey these important ideas and problems through concrete examples. We will study the Weil conjectures for varieties over finite fields, connections with character sums, construction of global zeta functions, and if time permits and depending on the interests/background of the audience, modern conjectures about special values of L-functions. While the focus is on the arithmetic of the varieties, this class should be of interests to those who would like to learn the connection between number theory, algebraic geometry and representation theory.