Speaker: Daqing Wan, Professor of University of Califonia at Irvine.

Abstract: This mini-course aims to provide a motivating and essentially self-contained introduction to zeta functions in number theory and arithmetic algebraic geometry, at the level of advanced mathematics undergraduate students, leading up to some of the current research directions.

The zeta function is a generation function which counts the number of primes, the number of rational points or the number of closed points on a system of polynomial equations over the integers. It contains deep information on the arithmetic, geometry and topology of the system of equations. Understanding the zeta function and its computation leads to many of the most profound results and the most difficult open problems in mathematics and computer science.

The first part is an introduction to the (Hasse-Weil) zeta function of an affine scheme of finite type over the integers (a finitely generated commutative algebra over the integers). We will give many examples, including the Riemann zeta function, the Dedekind zeta function, analytic continuation, generalized Riemann hypothesis, the Artin conjecture and its connection to modular forms.

The second part is an introduction to zeta functions over finite fields (i.e., when the scheme or variety has positive characteristic). This includes the celebrated Weil conjectures, Dwork's rationality theorem and Deligine's theorem on the Riemann hypothesis over finite fields.

The third part is an introduction to the p-adic absolute value (or Newton slopes) of the zeros of zeta functions over finite fields. We will focus on the case of toric hypersurfaces and describe its connection with toric geometry and Hodge theory. This can be viewed as the p-adic Riemann hypothesis for this zeta function.

The fourth part is an introduction to the variation of the Newton slopes when the toric hypersurface moves in an algebraic family. We will describe several decomposition theorems to compute the generic Newton slopes, based on convex triangulations of the polytopes. In the special case of a pair of reflexive polytopes, this leads to a possible arithmetic mirror conjecture for the corresponding mirror families of Calabi-Yau varieties.

Time: 2-5pm, Dec 24, 26, 31

Location: 82J04