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

We study the distribution of the numbers of \({F_{{q^r}}}\)-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 \({F_{{q^r}}}\)-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.