  The statistics of low-lying zeros of quadratic Dirichlet L-functions were
conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues
from the unitary symplectic ensemble. The n-level densities were found to be in
agreement with this in a certain neighborhood of the origin in the Fourier
domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the
neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH
gave the density as a complicated combinatorial factor, but it remained open
whether it coincides with the Random Matrix Theory factor. For n at most 7 this
was recently confirmed by Levinson and Miller. We resolve this problem for all
n, not by directly doing the combinatorics, but by passing to a function field
analogue, of L-functions associated to hyper-elliptic curves of given genus g
over a field of q elements. We show that the answer in this case coincides with
Gao's combinatorial factor up to a controlled error. We then take the limit of
large finite field size q to infinity and use the Katz-Sarnak equidistribution
theorem, which identifies the monodromy of the Frobenius conjugacy classes for
the hyperelliptic ensemble with the group USp(2g). Further taking the limit of
large genus g to infinity allows us to identify Gao's combinatorial factor with
the RMT answer.
