Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves

DOI10.1090/S0025-5718-09-02253-4zbMath1209.11059OpenAlexW2003033677WikidataQ123327504 ScholiaQ123327504MaRDI QIDQ3055205

Stefan Patrikis, Andrei Jorza, Corina E. Tarnita, William A. Stein, Grigor Grigorov

Publication date: 7 November 2010

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0025-5718-09-02253-4

zbMATH Keywords

Birch and Swinnerton-Dyer conjectureElliptic curves

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) (L)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture (11G40) Analytic computations (11Y35)

Related Items

Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture (Part II) ⋮ Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula ⋮ Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces ⋮ Examples of non-simple abelian surfaces over the rationals with non-square order Tate-Shafarevich group ⋮ Kolyvagin's result on the vanishing of \(\Sha(E/K)[p^\infty\) and its consequences for anticyclotomic Iwasawa theory] ⋮ Heegner points and the arithmetic of elliptic curves over ring class extensions ⋮ Visualizing elements of order in the Tate–Shafarevich group of an elliptic curve ⋮ Numerical modular symbols for elliptic curves ⋮ Lagrangian 4-planes in holomorphic symplectic varieties of \(K3^{[4}\)-type] ⋮ A $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties ⋮ Algorithms for the arithmetic of elliptic curves using Iwasawa theory ⋮ Algorithms for Chow-Heegner points via iterated integrals ⋮ Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups ⋮ Vanishing of Some Galois Cohomology Groups for Elliptic Curves

Uses Software

• Magma
• SageMath
• ecdata

Cites Work

• Height difference bounds for elliptic curves over number fields
• Heegner points and derivatives of \(L\)-series
• Mean values of derivatives of modular \(L\)-series
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• Modular elliptic curves and Fermat's Last Theorem
• The Cassels-Tate pairing on polarized Abelian varieties
• Galois properties of points of finite order of elliptic curves
• Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves
• Nonvanishing theorems for L-functions of modular forms and their derivatives
• On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises
• Arithmetic on Curves of Genus 1 : III. The Tate-Šafarevič and Selmer Groups
• Kolyvagin systems
• Visualizing Elements in the Shafarevich—Tate Group
• There are genus one curves over \mathbb{Q} of every odd index
• On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves
• Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero
• Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.
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