Number Theory Seminar

Wednesday 2/26/03 4:30-5:30 PM in Hylan 1106b:
Doug Ravenel- "The Weil Conjectures"

ABSTRACT:

This talk is a sequel to the SUMS talk of February 19 in which I explained the subject of the Weil conjectures. For an algebraic variety X in characteristic p, one wants to count the number of points defined over the field Fpn with pn elements for each n. This information is encoded in the zeta function of X, which is what the Weil conjectures are about. One can also consider the (infinite) set of points X(K) defined over the algebraic closure K of Fp. Fpn is the subfield of K fixed by a certain field automorphism. There is a similar automorphism of X(K) for which X(Fpn) is the fixed point set. If X(K) were a topological space, one could get information about the subspace fixed by an automorphism with the help of the Lefschetz Fixed Point Theorem, which I will explain in the talk. Weil's program to prove his conjectures was to find an analog to this topological method in algebraic geometry in characteristic p.