2011/07/31 by E Dueñez, Eduardo Dueñez, D K Huynh +7 · 14 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Characteristic polynomial #Distribution (mathematics) #Eigenvalues and eigenvectors #Elliptic curve #Geometry and complex manifolds #Matrix (chemical analysis) #Orthogonal matrix #Orthogonal polynomials #Random matrix #Scaling #math-ph #math.MP #math.NT #math.PR #msc:11G05 #msc:11G40 #msc:11M26 #msc:15B10 #msc:15B52
paper · pdf · doi:10.1088/1751-8113/45/11/115207
published in Journal of Physics A Mathematical and Theoretical 45(11), 115207 (Institute of Physics) · 38 pages, version 2 (added some plots)
arxiv created 2011/12/07 · openalex publication_date 2012/03/01 · arxiv updated 2015/03/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 centre of the critical strip was observed numerically by Miller (2006 Exp. Math. 15 257–79); 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 (2 N ) 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 modelling 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 (2 N ). 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.