Date: November 19, 2010
Speaker: Hiren Maharaj (Clemson)
Title: Modular curves and codes
Abstract: Let nt(2f)20101119/latex_610105a10417f439cb9add44f141f8fd736501cf_p1.png "$A(q)$")
denote the upper limit point of the ratio of the number of rational places of an algebraic function field (of a single variable) with finite field nt(2f)20101119/latex_e305ea603d091edcd9f41b87d2df472747895942_p1.png "$GF(q)$")
as the full field of constants to the genus. An increasing tower of function fields attaining this limit is said to be asymptotically optimal. Motivated by practical applications, coding theorists (Garcia and Stichtenoth) were the first to explicitly recursively construct such towers in the case that nt(2f)20101119/latex_6e580f4531cacbba0fa8c95500af1c93efd2bbd6_p1.png "$q$")
is a square. Elkies then demonstrated that every such tower they found came from modular curves and conjectures that if is a square, then all asymptotically optimal towers come from modular curves. I will talk about the modular connection and some of the problems I am interested in.