The statistics of the number of rational points in the hyperelliptic ensemble over a finite field

Abstract

We study the distribution of the numbers of -rational points of hyperelliptic curves over a finite field F (q) in odd characteristic. This extends the result of Kurlberg and Rudnick [4]. We also study the distribution of the number of -rational points and the trace of high powers of the Frobenius class of real hyperelliptic curves over a finite field F (q) in even characteristic