On the low-lying zeros of Hasse–Weil L-functions for elliptic curves
Copyright policy )On the low-lying zeros of Hasse–Weil L-functions for elliptic curves
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.
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
- UNSW Sydney Australia
- Nanyang Technological University Singapore
- University of East Anglia United Kingdom
- Jacobs University Germany
Mathematics(all), anzsrc-for: 4901 Applied mathematics, anzsrc-for: 4902 Mathematical physics, low-lying zeros, anzsrc-for: 0101 Pure Mathematics, Estimates on character sums, anzsrc-for: 4904 Pure Mathematics, 510, Elliptic curve L-functions, ranks of elliptic curves, 4903 Numerical and Computational Mathematics, Low-lying zeros, FOS: Mathematics, :Science::Mathematics::Number theory [DRNTU], Number Theory (math.NT), Other Dirichlet series and zeta functions, elliptic curve \(L\)-functions, 11M06, 11M26, 11M41, 11F30, 11G05, 11G40, 11L20, 11L40, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Mathematics - Number Theory, Ranks of elliptic curves, anzsrc-for: 4903 Numerical and Computational Mathematics, 4904 Pure Mathematics, anzsrc-for: 49 Mathematical Sciences, Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses, Elliptic curves over global fields, 49 Mathematical Sciences, Sums over primes
Mathematics(all), anzsrc-for: 4901 Applied mathematics, anzsrc-for: 4902 Mathematical physics, low-lying zeros, anzsrc-for: 0101 Pure Mathematics, Estimates on character sums, anzsrc-for: 4904 Pure Mathematics, 510, Elliptic curve L-functions, ranks of elliptic curves, 4903 Numerical and Computational Mathematics, Low-lying zeros, FOS: Mathematics, :Science::Mathematics::Number theory [DRNTU], Number Theory (math.NT), Other Dirichlet series and zeta functions, elliptic curve \(L\)-functions, 11M06, 11M26, 11M41, 11F30, 11G05, 11G40, 11L20, 11L40, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Mathematics - Number Theory, Ranks of elliptic curves, anzsrc-for: 4903 Numerical and Computational Mathematics, 4904 Pure Mathematics, anzsrc-for: 49 Mathematical Sciences, Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses, Elliptic curves over global fields, 49 Mathematical Sciences, Sums over primes
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