Traces of High Powers of the Frobenius Class in the Hyperelliptic Ensemble
نویسنده
چکیده
The Riemann Hypothesis, proved by Weil [19], is that the zeros of P (u) all lie on the circle |u| = 1/√q. Thus one may give a spectral interpretation of PC(u) as the characteristic polynomial of a 2g × 2g unitary matrix ΘC : PC(u) = det(I − u √ q ΘC) so that the eigenvalues eiθj of ΘC correspond to zeros q−1/2e−iθj of ZC(u). The matrix (or rather the conjugacy class) ΘC is called the unitarized Frobenius class of C. We would like to study how the Frobenius classes ΘC change as we vary the curve over a family of hyperelliptic curves of genus g, in the limit of
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