Low-lying zeros in families of elliptic curve L-functions over function fields

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions

SATO-TATE EQUIDISTRIBUTION OF CERTAIN FAMILIES OF ARTIN L-FUNCTIONS

We study various families of Artin L-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros

LOW-LYING ZEROS IN FAMILIES OF HOLOMORPHIC CUSP FORMS: THE WEIGHT ASPECT

We study low-lying zeros of L-functions attached to holomorphic cusp forms of level 1 and large even weight. In this family, the Katz-Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions phi satisfying the condition supp((phi) over cap) subset of (-2, 2). We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of (phi) over cap reaches the point 1. In particular, the first of these terms involves the quantity (phi) over cap (1) which appeared in the previous work of Fouvry-Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of the Generalized Riemann Hypothesis (GRH) for higher-degree automorphic L-functions. Finally, when supp((phi) over cap) subset of (-1, 1) we obtain an unconditional estimate which is significantly more precise than the prediction of the L-functions ratios conjecture

On the generalized circle problem for a random lattice in large dimension

In this note we study the error term R-n,R-L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f : Z(+) -> R+ satisfying lim(n ->infinity) f (n) = infinity and f (n) = O-epsilon(e(epsilon n)) for every epsilon > 0. Then, the random function t bar right arrow 1/root 2f (n) R-n,R-L (t f(n)) on the interval [0, 1] converges in distribution to one-dimensional Brownian motion as n -> infinity. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers\u27 mean value formula from [18]. For the individual kth moment of the variable (2f (n))(-1/2) R-n,R-L (f (n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f (n) = O(e(cn)) for any fixed c is an element of (0, c(k)), where c(k) is a constant depending on k whose optimal value we determine. (C) 2019 Elsevier Inc. All rights reserved

Non-vanishing of maass form L-functions at the central point

In this paper, we consider the family {Lj(s)}∞j=1 of L-functions associated to an orthonormal basis {uj}∞j=1 of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κ2j + 1/4}∞j=1. We prove the following effective non-vanishing result: At least 50% of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals