Pär Kurlberg: Point count statistics for families of curves over finite fields - random matrix theory vs positivity

We investigate the distribution of the number of F_p-points of curves in various families, where F_p is the finite field with p elements. If we consider a family of curves having fixed genus g and let p tend to infinity the situation is fairly well understood - the distribution of the point count fluctuations are given by generalized Sato-Tate distributions, which in turn is closely related to random matrix theory. On the other hand, if p is fixed and we let g tend to infinity (or taking p, g to infinity in some arbitrary way), the situation is less clear, e.g., since the number of points on a curve cannot be negative, the random matrix theory model is not valid in this setting. However, for some families of curves, certain "coin flip models" (closely related to character sums in the case of hyperelliptic curves) can be used to describe the fluctuations; using this we can show that the point count fluctuations are Gaussian in the large genus limit.