We investigate the large weight (k → ∞) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of L-functions,here φ is a fixed even Hecke-Maass cusp form and H k is a Hecke eigenbasis for the space H k of holomorphic cusp forms of weight k for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1-and 2-level densities, we find evidence of underlying symplectic and SO(even) symmetry, respectively. This should be contrasted with previous results of Iwaniec-Luo-Sarnak for the families {L(s, f ) : f ∈ H k } and {L(s, sym 2 f ) : f ∈ H k }, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus, the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.

Let {FN } and {GM } be families of primitive automorphic L-functions for GLn(A Q ) and GLm(A Q ), respectively, such that, as N, M → ∞, the statistical behavior (1-level density) of the low-lying zeros of L-functions in FN and GM agrees with that of the eigenvalues near 1 of matrices in G1 and G2, respectively, as the size of the matrices tend to infinity, where each Gi is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions FN × GM are automorphic, we study their 1-level density. (We also study convolved families of the form f × GM for a fixed f .) Under natural assumptions on the families (which hold in many cases), we can associate to each family L of L-functions a symmetry constant cL equal to 0, 1, or −1 if the corresponding low-lying zero statistics agree with those of the unitary symplectic, or orthogonal group, respectively. Our main result is that cF×G = cF · cG: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f × GM . We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with positive rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M → ∞, as lower-order terms).

We introduce random matrix ensembles that correspond to the infinite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families of (unitary, orthogonal and symplectic) ensembles of Jacobi type. We discuss the universal and weakly universal features of the global and local correlations of the levels in the bulk and at the "hard" edge of the spectrum (i. e., at the "central points" ±1 on the unit circle). Previously known results are extended, and we find new simple formulas for the Bessel Kernels that describe the local correlations at a hard edge.

Abstract. We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random matrix model for Miller's surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this sub-ensemble of SO(2N ) the excised orthogonal ensemble. The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of L-functions implied by the formulae of Waldspurger and Kohnen-Zagier. The cut-off scale appropriate to modeling elliptic curve L-functions is exponentially small relative to the matrix size N . The one-level density of the excised ensemble can be expressed in terms of that of the well-known Jacobi ensemble, enabling the former to be explicitly calculated. It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of SO(2N ). When N → ∞ we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller's discrepancy. Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data.Date: December 8, 2011. 2010 Mathematics Subject Classification. 11M26, 15B52 (primary), 11G05, 11G40, 15B10 (secondary).