Low-lying zeros of families of elliptic curves

2004/06/30 by Matthew P. Young, Matthew Young · 9 citations
Mathematics · #Analytic Number Theory Research #Mathematical functions and polynomials #Random Matrices and Applications #math.NT #msc:11M26

paper · pdf · doi:10.1090/s0894-0347-05-00503-5

v2: Enhanced exposition, 56 pages. v3: One reference added and one sentence changed in the paragraph following Corollary 3.4

arxiv created 2005/04/06 · openalex publication_date 2005/09/07 · openalex created_date 2016/06/24 · arxiv updated 2020/08/17 · openalex updated_date 2026/07/28

Abstract

We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two, which implies that a positive proportion of elliptic curves over the rationals have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. These results are conditional on the Generalized Riemann Hypothesis.

Citations

• The average rank of elliptic curves I
• Modular Elliptic Curves and Fermat's Last Theorem

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• A random matrix model for elliptic curveL-functions of finite conductor
• Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
• One level density of low-lying zeros of quadratic Hecke L-functions of Imaginary Quadratic Number Fields
• Murmurations for elliptic curves ordered by height
• On the counting function of cubic function fields
• Rank of an elliptic curve [wikipedia]