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Type of Publication:  Journal article 
Author:  Huynh, Duc K.; Keating, Jon P.; Snaith, Nina C. 
Year of publication:  2009 
Published in:  Journal of Number Theory ; 129 (2009), 12.  pp. 28832902.  ISSN 0022314X.  eISSN 10961658 
DOI (citable link):  https://dx.doi.org/10.1016/j.jnt.2008.12.008 
Summary: 
It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve Lfunctions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one and twolevel densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the Lfunctions ratios conjectures to calculate the 1level density for the family of even quadratic twists of an elliptic curve Lfunction for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.

Subject (DDC):  510 Mathematics 
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HUYNH, Duc K., Jon P. KEATING, Nina C. SNAITH, 2009. Lower Order Terms for the OneLevel Density of Elliptic Curve LFunctions. In: Journal of Number Theory. 129(12), pp. 28832902. ISSN 0022314X. eISSN 10961658. Available under: doi: 10.1016/j.jnt.2008.12.008
@article{Huynh2009Lower25392, title={Lower Order Terms for the OneLevel Density of Elliptic Curve LFunctions}, year={2009}, doi={10.1016/j.jnt.2008.12.008}, number={12}, volume={129}, issn={0022314X}, journal={Journal of Number Theory}, pages={28832902}, author={Huynh, Duc K. and Keating, Jon P. and Snaith, Nina C.} }
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