A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor

• Published in: arXiv.org
• Main Authors: Dueñez, Eduardo, Huynh, Duc Khiem, Keating, Jon P, Miller, Steven J, Snaith, Nina C
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 07.12.2011
• Subjects: Conductors; Constraining; Curves; Density; Eigenvalues; Mathematical analysis; Mathematical models; Mathematics - Mathematical Physics; Mathematics - Number Theory; Mathematics - Probability; Physics - Mathematical Physics; Polynomials; Rescaling
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1107.4426

We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; 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 formula 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 tends to infinity 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.