An elliptic curve test of the L-Functions Ratios Conjecture

We compare the L-Function Ratios Conjecture's prediction with number theory for the family of quadratic twists of a fixed elliptic curve with prime conductor, and show agreement in the 1-level density up to an error term of size X^{-(1-sigma)/2} for test functions supported in (-sigma, sigma); this gives us a power-savings for \sigma<1. This test of the Ratios Conjecture introduces complications not seen in previous cases (due to the level of the elliptic curve). Further, the results here are one of the key ingredients in the companion paper [DHKMS2], where they are used to determine the effective matrix size for modeling zeros near the central point for this family. The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining the data observed by Miller in [Mil3]. A key ingredient in our analysis is a generalization of Jutila's bound for sums of quadratic characters with the additional restriction that the fundamental discriminant be congruent to a non-zero square modulo a square-free integer M. This bound is needed for two purposes. The first is to analyze the terms in the explicit formula corresponding to characters raised to an odd power. The second is to determine the main term in the 1-level density of quadratic twists of a fixed form on GL_n. Such an analysis was performed by Rubinstein [Rub], who implicitly assumed that Jutila's bound held with the additional restriction on the fundamental discriminants; in this paper we show that assumption is justified.Comment: 35 pages, version 1.2. To appear in the Journal of Number Theor

Lower order terms for the moments of symplectic and orthogonal families of LL-functions

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an elliptic curve over \Q. In so doing, we are led to study determinants of binomial coefficients of the form $\det (\binom{2k-i-\lambda_{k-i+1}}{2k-2j})$.Comment: 34 pages, 4 table

Elliptic curve L-functions of finite conductor and random matrix theory

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One-level density of families of elliptic curves and the Ratios Conjecture

Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The one-level scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the one-level scaling density for the second family is the sum of the Dirac distribution and the even orthogonal distribution. This is a new phenomenon for a family of curves with odd rank: the trivial zero at the central point accounts for the Dirac distribution, and also affects the remaining part of the scaling density which is then (maybe surprisingly) the even orthogonal distribution. The one-level density for this family was studied in the past for test functions with Fourier transforms of limited support, but since the Fourier transforms of the even orthogonal and odd orthogonal distributions are undistinguishable for small support, it was not possible to identify the distribution with those techniques. This can be done with the Ratios Conjecture, and it sheds more light on “independent” and “non-independent” zeroes, and the repulsion phenomenon.publishe

Lower order terms for the moments of symplectic and orthogonal families of L-functions, J. Number Theory 133 (2013) 639—674; arXiv:1203.4647; MR2994379. Dirichlet L-Series 17

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic and also of the L-functions associated to quadratic twists of an elliptic curve over Q. In so doing, we are led to study determinants of binomial coefficients of the form det 2k−2 j