Seminars & Colloquia Calendar

ABSTRACT:  It is a remarkable fact that the Riemann zeta function extends to a meromorphic function on the whole complex plane. A conjecture of Weil, proved by Dwork, asserts that the zeta function of any variety over a finite field is likewise meromorphic, from which it follows that it can be expressed as a rational function. In the case of curves, Kapranov observed that this is true in a very strong sense, which continues to hold even in characteristic zero. He asked whether this remains true for higher dimensional varieties. Valery Lunts and I disproved his conjecture fifteen years ago, and recently disproved a weaker conjecture due to Denef and Loeser. This explains, in some sense, why Weil's conjecture was so much easier in dimension 1 than in higher dimension.