Type of Publication:  Journal article 
Publication status:  Published 
URI (citable link):  http://nbnresolving.de/urn:nbn:de:bsz:35225vsatko2gb0l1 
Author:  David, Chantal; Huynh, Duc Khiem; Parks, James 
Year of publication:  2015 
Published in:  Research in Number Theory ; 1 (2015), 1.  6.  Springer.  ISSN 25220160.  eISSN 23639555 
DOI (citable link):  https://dx.doi.org/10.1007/s4099301500057 
Summary: 
Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the onelevel density for two families of Lfunctions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The onelevel scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the onelevel 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 onelevel 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 “nonindependent” zeroes, and the repulsion phenomenon.

Subject (DDC):  510 Mathematics 
Keywords:  Number Theory, Elliptic Curve, Elliptic Curf, Chebyshev Polynomial, Root Number 
Link to License:  Attribution 4.0 International 
Bibliography of Konstanz:  Yes 
Refereed:  Unknown 
DAVID, Chantal, Duc Khiem HUYNH, James PARKS, 2015. Onelevel density of families of elliptic curves and the Ratios Conjecture. In: Research in Number Theory. Springer. 1(1), 6. ISSN 25220160. eISSN 23639555. Available under: doi: 10.1007/s4099301500057
@article{David201512Onele53232, title={Onelevel density of families of elliptic curves and the Ratios Conjecture}, year={2015}, doi={10.1007/s4099301500057}, number={1}, volume={1}, issn={25220160}, journal={Research in Number Theory}, author={David, Chantal and Huynh, Duc Khiem and Parks, James}, note={Article Number: 6} }
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