Date & Time: Wednesday, January 21, 2015, 16:00-17:00.
Venue: Ramanujan Hall
Speaker: Peter Beelen, Technical University of Denmark
Title: Curves with many rational points: the latest on Ihara's constant
Abstract: Given a finite field GF(q), one may ask how many rational points (i.e. points defined over GF(q) ) an algebraic curve may have. Given a non-singular, absolutely irreducible and projective algebraic curve C, this quantity is usually denoted by N_1(C). Hasse and Weil showed that N_1(C) is at most q+1+2sqrt(q) g(C), with g(C) the genus of C. Ihara was interested in the maximal value of the quantity A(q)= limsup N_1(C)/g(C), where the limsup is taken over all families of algebraic curves as above with genus tending to infinity. The quantity A(q) is now often called Ihara's constant.
Drinfeld and Vladut showed that A(q) is at most sqrt(q)-1, while Ihara himself showed that A(q) is at least sqrt(q)-1 in case q is a square. For the nonsquare case it is still open what the exact value of A(q) is. In this talk, I will discuss recent progress (joint work with A. Bassa, A. Garcia, and H. Stichtenoth) in which a good lower bound for A(q) was obtained in case q is not a prime. Equations for algebraic curves will be obtained using the theory of Drinfeld modules and using these a sketch of the proof of the following fact will be given.
Let q=p^(2n+1), then A(q) is bounded from below by the harmonic mean of p^(n+1)-1 and p^n-1.
This lower bound is currently the best known lower bound for A(q) in case q=p^(2n+1) for n >0.