The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions

We investigate the large weight (k --> oo) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s,phi x f): f in H_k(1)} and {L(s,phi times sym^2 f): f in H_k(1)}; here phi is a fixed even Hecke-Maass cusp form and H_k(1) is a Hecke eigenbasis for the space H_k(1) 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 in H_k(1)} and {L(s,sym^2f): f in H_k(1)}, 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.Comment: 26 pages: revised draft: fixed some typos, added an appendix with the calculations for the signs of the functional equation and Gamma factor

The effect of convolving families of L-functions on the underlying group symmetries

L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M --> oo, the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary, symplectic or orthogonal). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M 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 c_L equal to 0 (resp., 1 or -1) if the corresponding low-lying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: 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 x G_M. 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 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 --> oo, as lower-order terms).Comment: 41 pages, version 2.1, cleaned up some of the text and weakened slightly some of the conditions in the main theorem, fixed a typ

Metastable convergence of ergodic averages: The continuous logic viewpoint.

We revisit certain classical and recent results on convergence of averages of a fixed element f of a topological vector space V endowed with an action (g,f)â ¦ áµ f of an amenable (semi)group G. (In the special case when G = â is the semigroup of naturals, the averages are just (¹f + ²f + â ¯ + â ¿f)/n). Such results, collectively called ergodic convergence theoremsâ although there is really nothing â ergodicâ about themâ , include the classical ergodic theorem of Birkhoff as well as von Neumannâ s mean ergodic theorem (MET), alongside subsequent generalizations. In collaboration with J. Iovino, we use continuous logic to obtain a radically elementary proof of a MET valid for any polynomial action of an amenable group on a Hilbert space. The Compactness Theorem from logic implies the existence of universal rates of metastable convergence that depend only on the degree of the action.Non UBCUnreviewedAuthor affiliation: Spelman CollegeResearche