Date & Time: Wednesday, December 23, 2009, 16:00-17:00
Venue: Ramanujan Hall
Title: On Towers over Finite Fields
Speaker: Prof. Arnaldo Garcia Instituto Nacional de Matem`tica Pura e Aplicada (IMPA) a Rio de Janeiro, Brazil
Abstract: A tower of curves over a finite field k is an infinite sequence of curves Cm and of maps Cm → Cm−1 defined over k, with genus g(Cm ) growing to infinity. A tower has a funda- mental invariant called its limit that measures the asymptotic behaviour of the ratios of the number of rational points of Cm by the genus g(Cm ). A linear code over k is just a subspace of the vector space k n . We shall motivate the study of towers by the connection to the asymptotic theory of codes. For this application to Coding Theory, the tower should be explicit; i.e., the curves Cm are explicitly given by polynomial equations with coefficients in the finite field k. We shall survey some explicit constructions of towers over finite fields obtained together with H. Stichtenoth, and relate them with results by Ihara, Serre, Geer-Vlugt, Zink. This talk should be accessible to a general audience.