Seminars on Algebraic Curves over Finite Fields and Coding Theory

Speaker: Dr. Arnaldo Garcia

Researcher at Instituto Nacional de Matematica Pura e Aplicada, IMPA

Seminar  1:  On Algebraic Curves Over Finite Fields

Date: February 27, 2012, Monday, 10:30
Place: Department of Mathematics, Room 203

Bounding the number of points, with coordinates in a finite field, on algebraic curves has attracted much attention, especially after the discovery of V.D. Goppa of good linear codes from such algebraic curves. This talk will be a survey on Curves with Many Points, specially the so-called maximal curves; i.e., the ones attaining Hasse-Weil upper bound (equivalent to the validity of Riemann Hypothesis in this context).

Seminar  2: Explicit Towers Over Non-Prime Finite Fields

Date: February 29, 2012, Wednesday, 15:30
Place: Institute of Applied Mathematics, S-209


Bounding the number of rational points (rational places) on algebraic curves (function fields) over finite fields has attracted much attention. The most famous result here is the Hasse-Weil bound which is equivalent to the Riemann Hypothesis in this context. Ihara was the first to realize that the Hasse-Weil upper bound becomes weaker as the genus grows.  He then introduced a quantity (now known as Ihara’s quantity) that controls the asymptotic on rational points (rational places) as the genus grows to infinity. The only situation where the exact value of this quantity is known is the case of finite fields of square cardinalities (due to Ihara using Shimura modular curves). Over finite fields of cubic cardinalities one has a good lower bound (due to Zink and Bezerra-Garcia-Stichtenoth). For any other cardinality (not a square or a cube) essentially nothing was known about the behaviour of Ihara’s quantity. The aim of this talk is to present some explicit infinite towers of curves (function fields) giving very good lower bounds for this quantity over any non-prime finite field. This is joint work with Bassa-Beelen-Stichtenoth.

Institute of Applied Mathematics,
Society for Industrial and Applied Mathematics,
Department of Mathematics,

Speaker’s visit to Turkey is supported by Tubitak.