Date & Time: Wednesday, December 23, 2009, 11:00-12:00

Venue: Ramanujan Hall

Title: On Maximal Curves

Speaker: Prof. Arnaldo Garcia Instituto Nacional de Matem`tica Pura e Aplicada (IMPA) Rio de Janeiro, Brazil

Abstract: Curves over finite fields k of square cardinality q 2 attaining the Hasse-Weil upper bound for the number of rational points over k are called maximal curves. Hasse-Weil bound is equivalent to the validity of the Riemman Hypothesis in this context of curves over finite fields. The Hermitian curve over k (which can be represented as the Fermat curve of degree q + 1) is the unique maximal curve over k with the biggest genus possible and, by a result of Serre, curves covered by the Hermitian are also maximal curves over k. A natural question: Is any maximal curve over k covered by the Hermitian curve? This question has a negative answer obtained recently by Giulietti-Korchm ́ros. The aim of this a talk is to survey maximal curves, and also to present the curve of Giulietti-Korchm ́ros a and generalizations of it obtained together with G ̈neri and Stichtenoth. We will also point u out that the Hermitian curve has exceptional behaviour with respect to Weierstrass point theory, Galois point theory, automorphism group, Fr ̈benius orders, etc. o