The fluctuations in the number of points on a hyperelliptic curve over a finite field

The number of points on a hyperelliptic curve over a field of q elements may be expressed as q + 1 + S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that S / sqrt(q) is distributed as the trace of a random 2 g × 2 g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values ±1 with probabilities 1 / 2 (1 + q^-1) and the value 0 with probability 1 / (q + 1). When both the genus and the finite field grow, we find that S / sqrt(q) has a standard Gaussian distribution.