Matt Baker

``Counting Points on $X_0(l)$ over $F_{p^2}$''

April 14, 1999

Modular curves have become popular in coding theory because certain families over certain finite fields attain the "Vladut-Drinfeld" bound. In this talk we will begin by discussing the bound of Vladut-Drinfeld, and then show that the family of modular curves X_0(l) attain this bound as l goes to infinity. This is well-known. We also will prove the following more precise fact, whose statement has appeared in the literature, but never accompanied by a proof: if l is a prime not equal to p, and g denotes the genus of X_0(l), then the absolute value of the difference between (p-1)g and the number of points of X_0(l) over F_{p^2} is at most C_p, a constant which depends only on p. Our proof of this fact is based on email discussions with Mike Zieve.