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 is distributed as the trace of a random 2g×2g 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 has a standard Gaussian distribution.