2007/08/31 by Stephan Baier, Liangyi Zhao · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic number #Analytic Number Theory Research #Combinatorics #Elliptic curve #Mathematical analysis #Mathematics #Pure mathematics #Rank (graph theory) #Riemann hypothesis #Supersingular elliptic curve #math.NT #msc:11F30 #msc:11G05 #msc:11G40 #msc:11L20 #msc:11L40 #msc:11M06 #msc:11M26 #msc:11M41
paper · pdf · doi:10.1016/j.aim.2008.06.006
published as Adv. Math., Vol. 219, No. 3, 2008, pp. 952-985. · 23 pages. The majorant of the average analytic rank of all elliptic curves have been improved to 27/14 instead of the 241/122 in the previous version of the paper. To appear in Adv. Math
arxiv created 2008/06/15 · openalex publication_date 2008/07/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevic group. Statements of this flavor were known previously under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.