Low-lying zeros of elliptic curve <i>L</i>-functions: Beyond the Ratios Conjecture

Daniel Fiorilli, James Parks, Anders Södergren

Type: Article

Publication Date: 2016-01-08

Citations: 12

DOI: https://doi.org/10.1017/s0305004115000730

Abstract

Abstract We study the low-lying zeros of L -functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$ . We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L -functions Ratios Conjecture.

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Mathematical Proceedings of the Cambridge Philosophical Society - View
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Action	Title	Year	Authors
+	Rang moyen de familles de courbes elliptiques et lois de Sato-Tate	1995	Philippe Michel
+ PDF Chat	The Arithmetic of Elliptic Curves	1986	Joseph H. Silverman
+	Multiplicative Number Theory	1980	H. Davenport
+	Conjectures on elliptic curves over quadratic fields	1979	Dorian Goldfeld
+	Algorithms for Modular Elliptic Curves	1992	J. E. Cremona
+	Random Matrices, Frobenius Eigenvalues, and Monodromy	1998	Nicholas M. Katz Peter Sarnak
+ PDF Chat	Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves	2005	Matthew P. Young
+	The n-level correlations of zeros of the zeta function	1994	Zeév Rudnick Peter Sarnak
+	On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises	2001	Christophe Breuil Brian Conrad Fred Diamond Richard Taylor
+	Random Matrix Theory and L-Functions at s = 1/2	2000	Jonathan P. Keating N. C. Snaith