  We study the averages of products of traces of high powers of the Frobenius
class of hyperelliptic curves of genus g over a fixed finite field. We show
that for increasing genus g, the limiting expectation of these products equals
to the expectation when the curve varies over the unitary symplectic group
USp(2g). We also consider the scaling limit of linear statistics for
eigenphases of the Frobenius class of hyperelliptic curves, and show that their
first few moments are Gaussian.
