Mean Values of L-Functions for the Hyperelliptic Ensemble

Thanks to the work of Katz and Sarnak on L-functions over function fields, we know that the Frobenius classes associated to L-functions of hyperelliptic curves over a finite field with $q$ elements, $F_{q}$, becomes equidistributed in the unitary symplectic group in the limit as the genus of the curve is fixed and $q$ is large. In this talk we review this and related results about L-functions associated to hyperelliptic curves over the rational function field, specifically we present some results about moments of L-functions attached to an ensemble of Hyperelliptic curves in the opposite limit studied by Katz and Sarnak, we derive asymptotic formulas for the moments as $q$ is fix and the genus is large.