Abstract. For a positive integer m and a subgroup Λ of the unit group (Z/mZ) × , the corresponding generalized Kloosterman sum is the function). Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.
In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.
We introduce a new family of N × N random real symmetric matrix ensembles, the k-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but k eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as N → ∞; the remaining k are tightly constrained near N/k and their distribution converges to the k × k hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We isolate the two regimes by using matrix perturbation results and a nonstandard weight function for the eigenvalues, then derive their limiting distributions using a modification of the method of moments and analysis of the resulting combinatorics.
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