Many of the motivating questions of algebraic geometry center around solutions to a system of polynomial equations over an algebraically closed field. However, if the system has integer coefficients, we can just as well consider solutions in a finite field. In this case, a new question emerges: how many are there?

Given a finite field and its extensions, we can organize the numbers of points on a variety over those fields into a generating function called a zeta function of the variety. The Weil conjectures make the remarkable claim that, even though varieties over finite fields are only finite collections of points, this zeta function still carries information about the variety's geometry. Furthermore, they give elegant formulas and bounds for the point counts, much like how the properties of the Riemann zeta function relate to the distribution of primes. These conjectures motivated much of the 20th-century development of algebraic geometry, and were finally proven in the 60s and 70s by Grothendieck, Deligne, and others.

In this talk, I'll introduce the Weil conjectures in the special case of algebraic curves. First, I'll define the zeta function of a curve, outline its resemblance to the original Riemann zeta function, and show how the latter's functional equation and Riemann hypothesis motivate the corresponding Weil conjectures. Then, I'll show how to get concrete formulas for counting points on a given curve over a given finite field. Time permitting, I'll say a bit about a proof of the conjectures for curves due to Bombieri, and mention how the Weil conjectures for a general projective variety draw on its geometry and topology. Speaker(s): Will Dana (UM)