Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory

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 (n-level density of the low-lying zeros of quadratic Dirichlet L-functions, 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 ≤ 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 → ∞ 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→ ∞ allows us to identify Gao’s combinatorial factor with the RMT answer.