How to do a 𝑝-descent on an elliptic curve

DOI10.1090/S0002-9947-03-03366-XzbMath1119.11029MaRDI QIDQ4441751

Edward F. Schaefer, Michael Stoll

Publication date: 7 January 2004

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Arithmetic ground fields for curves (14H25) Elliptic curves (14H52) Arithmetic ground fields for abelian varieties (14K15)

Related Items (36)

The \(p\)-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large ⋮ Local solubility and height bounds for coverings of elliptic curves ⋮ Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula ⋮ Selmer groups as flat cohomology groups ⋮ Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces ⋮ Exhibiting SHA[2 on hyperelliptic Jacobians] ⋮ Visualising Sha[2 in Abelian surfaces] ⋮ Chabauty Without the Mordell-Weil Group ⋮ Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve ⋮ Twists of \(X(7)\) and primitive solutions to \(x^2+y^3=z^7\) ⋮ Watkins’s conjecture for elliptic curves with non-split multiplicative reduction ⋮ Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III ⋮ Extensions and torsors for finite group schemes ⋮ ON THE FIBRES OF AN ELLIPTIC SURFACE WHERE THE RANK DOES NOT JUMP ⋮ Examples of non-simple abelian surfaces over the rationals with non-square order Tate-Shafarevich group ⋮ Ranks, 2\-Selmer groups, and Tamagawa numbers of elliptic curves with \(\mathbb{Z} /2\mathbb{Z} \times \mathbb{Z} /8\mathbb{Z}\)-torsion ⋮ On 7-division fields of CM elliptic curves ⋮ Cube sum problem for integers having exactly two distinct prime factors ⋮ Potential \(\text Ш\) for abelian varieties ⋮ Computing the Cassels–Tate pairing on 3-isogeny Selmer groups via cubic norm equations ⋮ On the 2‐part of the Birch and Swinnerton‐Dyer conjecture for quadratic twists of elliptic curves ⋮ An explicit family of cubic number fields with large 2-rank of the class group ⋮ Explicit isogeny descent on elliptic curves ⋮ Descent via isogeny on elliptic curves with large rational torsion subgroups ⋮ Efficient Compression of SIDH Public Keys ⋮ Models of some genus one curves with applications to descent ⋮ Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton-Dyer conjecture ⋮ GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3 ⋮ Explicit n-descent on elliptic curves, I. Algebra ⋮ Fields generated by torsion points of elliptic curves ⋮ 3-Selmer groups for curves y 2 = x 3 + a ⋮ Algorithms for the arithmetic of elliptic curves using Iwasawa theory ⋮ Explicit $n$-descent on elliptic curves III. Algorithms ⋮ Second $p$-descents on elliptic curves ⋮ Selmer groups of elliptic curves that can be arbitrarily large. ⋮ Computing the Selmer group of an isogeny between Abelian varieties using a further isogeny to a Jacobian

Uses Software

• KANT/KASH
• Magma
• ecdata

Cites Work

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• Class numbers of quadratic fields and Shimura's correspondence
• Die Klassengruppen quadratischer und kubischer Zahlkörper und die Selmergruppen gewisser elliptischer Kurven
• The Magma algebra system. I: The user language
• KANT V4
• Computing a Selmer group of a Jacobian using functions on the curve
• Class groups and Selmer groups
• Finding rational points on bielliptic genus 2 curves
• Implementing 2-descent for Jacobians of hyperelliptic curves
• Explicit 4-descents on an elliptic curve
• Arithmetic on Curves of Genus 1. I. On a conjecture of Selmer.
• The selmer groups of elliptic curves and the ideal class groups of quadratic fields
• Second descents for elliptic curves
• Computing the $p$-Selmer group of an elliptic curve
• Visualizing Elements in the Shafarevich—Tate Group
• A formula for the Selmer group of a rational three-isogeny
• Computing the Rank of Elliptic Curves over Number Fields
• The Selmer groups and the ambiguous ideal class groups of cubic fields
• Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.

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