A Family of Semistable Elliptic Curves with Large Tate-Shafarevitch Groups

DOI10.2307/2045480zbMath0567.14018OpenAlexW4247003770MaRDI QIDQ3682598

Kenneth Kramer

Publication date: 1983

Full work available at URL: https://doi.org/10.2307/2045480

zbMATH Keywords

Selmer groupBirch-Swinnerton-Dyer conjectureFamilies of elliptic curves with large Tate-Shafarevich groups

Mathematics Subject Classification ID

Families, moduli of curves (algebraic) (14H10) Elliptic curves (14H52) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) (14G10)

Related Items (14)

The \(p\)-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large ⋮ Potential \(\text Ш\) for abelian varieties ⋮ Arbitrarily large Tate-Shafarevich group on abelian surfaces ⋮ Elements of class groups and Shafarevich-Tate groups of elliptic curves ⋮ Large Shafarevich-Tate groups over quadratic number fields ⋮ Distribution of Selmer groups of quadratic twists of a family of elliptic curves ⋮ Average size of $2$-Selmer groups of elliptic curves, I ⋮ On elliptic curves with large Tate-Shafarevich groups ⋮ Some examples of 5 and 7 descent for elliptic curves over \(\mathbb{Q}\) ⋮ Large Selmer groups over number fields ⋮ Arbitrarily large 2-torsion in Tate–Shafarevich groups of abelian varieties ⋮ Computing the rank of elliptic curves over real quadratic number fields of class number 1 ⋮ The Cassels-Tate pairing and the Platonic solids. ⋮ Selmer groups of elliptic curves that can be arbitrarily large.

Cites Work

• The arithmetic of elliptic curves
• The rank of elliptic curves
• Algorithm for determining the type of a singular fiber in an elliptic pencil
• Die Ordnung der Schafarewitsch‐Tate‐Gruppe kann beliebig groß werden
• Arithmetic on Curves of genus 1. VI. The Tate-Safarevic group can be arbitrarily large.
• Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.
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