Abstract

For a fixed q and any n ≥ 1, the number of F_{q^n} -points on a hyperelliptic curve over F_q of genus g can be written as a q^n +1+S, where S is a certain character sum. We show that S behaves as a sum of q^n + 1 independent random variables as g → ∞, with values depending on the parity of n. We get our result by generalizing the result of Kurlberg and Rudnick [1] for the distribution of the affine F_q-points to any finite extension F_{q^n} of F_q, and using the techniques of Bucur, David, Feigon, and Lalin [2] to also consider the points at infinity over the full space of of hyperelliptic curves of genus g.