An elliptic curve test of the L-Functions Ratios Conjecture

• Published in: Journal of number theory 2011-06, Vol.131 (6), p.1117-1147
• Main Authors: Huynh, Duc Khiem, Miller, Steven J., Morrison, Ralph
• Format: Article
• Language: English
• Subjects: 1-Level density; Dirichlet L-functions; Low lying zeros; Ratios Conjecture; Twists of an elliptic curve
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2010.12.004

We compare the L-Function Ratios Conjectureʼs prediction with number theory for quadratic twists of a fixed elliptic curve, showing agreement in the 1-level density up to O ( X − 1 − σ 2 ) for test functions supported in ( − σ , σ ) , giving a power-savings for σ < 1 . This test introduces complications not seen in previous cases (due to the level of the elliptic curve). The results here are a key ingredients in Dueñez et al. (preprint) [DHKMS2], which determine the effective matrix size for modeling zeros near the central point. The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining data observed in Miller (2006) [Mil3]. A key ingredient is generalizing Jutilaʼs bound for quadratic character sums restricted to fundamental discriminant congruent to non-zero squares modulo a square-free integer. Another application is determining the main term in the 1-level density of quadratic twists of a fixed GL n form; this generalization was implicitly assumed in Rubinstein (2001) [Rub]. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=-Cbj1n5y-WE.