**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