The statistics of low-lying zeros of quadratic Dirichlet L-functions wereconjectured by Katz and Sarnak to be given by the scaling limit of eigenvaluesfrom the unitary symplectic ensemble. The n-level densities were found to be inagreement with this in a certain neighborhood of the origin in the Fourierdomain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend theneighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRHgave the density as a complicated combinatorial factor, but it remained openwhether it coincides with the Random Matrix Theory factor. For n at most 7 thiswas recently confirmed by Levinson and Miller. We resolve this problem for alln, not by directly doing the combinatorics, but by passing to a function fieldanalogue, of L-functions associated to hyper-elliptic curves of given genus gover a field of q elements. We show that the answer in this case coincides withGao's combinatorial factor up to a controlled error. We then take the limit oflarge finite field size q to infinity and use the Katz-Sarnak equidistributiontheorem, which identifies the monodromy of the Frobenius conjugacy classes forthe hyperelliptic ensemble with the group USp(2g). Further taking the limit oflarge genus g to infinity allows us to identify Gao's combinatorial factor withthe RMT answer.
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