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

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DUEÑEZ, Eduardo, Duc HUYNH, Jon P. KEATING, Steven J. MILLER, Nina C. SNAITH, 2012. A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor. In: Journal of Physics A: Mathematical and Theoretical. 45(11), 115207. ISSN 1751-8113. eISSN 1751-8121

@article{Duenez2012Rando-25389, title={A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor}, year={2012}, doi={10.1088/1751-8113/45/11/115207}, number={11}, volume={45}, issn={1751-8113}, journal={Journal of Physics A: Mathematical and Theoretical}, author={Dueñez, Eduardo and Huynh, Duc and Keating, Jon P. and Miller, Steven J. and Snaith, Nina C.}, note={Article Number: 115207} }

<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > <rdf:Description rdf:about="https://kops.uni-konstanz.de/rdf/resource/123456789/25389"> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2013-12-13T16:13:07Z</dc:date> <dcterms:rights rdf:resource="http://nbn-resolving.org/urn:nbn:de:bsz:352-20140905103605204-4002607-1"/> <dc:creator>Keating, Jon P.</dc:creator> <dcterms:abstract xml:lang="eng">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(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 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(2N). 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.</dcterms:abstract> <dcterms:issued>2012</dcterms:issued> <dc:contributor>Huynh, Duc</dc:contributor> <dc:contributor>Dueñez, Eduardo</dc:contributor> <dc:rights>deposit-license</dc:rights> <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/25389"/> <dc:creator>Miller, Steven J.</dc:creator> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2013-12-13T16:13:07Z</dcterms:available> <dc:creator>Huynh, Duc</dc:creator> <dcterms:title>A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor</dcterms:title> <dc:contributor>Miller, Steven J.</dc:contributor> <dc:contributor>Keating, Jon P.</dc:contributor> <dc:creator>Snaith, Nina C.</dc:creator> <dcterms:bibliographicCitation>Journal of Physics A : Mathematical and Theoretical ; 45 (2012), 11. - 115207</dcterms:bibliographicCitation> <dc:language>eng</dc:language> <dc:creator>Dueñez, Eduardo</dc:creator> <dc:contributor>Snaith, Nina C.</dc:contributor> </rdf:Description> </rdf:RDF>

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