The distribution of rational points on curves over a finite field on average

Algebra Seminar
Monday, April 1, 2013 - 3:00pm
1 hour (actually 50 minutes)
Skiles 006
Georgia Tech
Let p be a prime, let C/F_p be an absolutely irreducible curve inside the affine plane. Identify the plane with D=[0,p-1]^2. In this talk, we consider the problem of how often a box B in D will contain the expected number of points. In particular, we give a lower bound on the volume of B that guarantees almost all translations of B contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense. This is joint work with Alexandru Zaharescu.