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On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

Eleni Agathocleous

Abstract

We study an infinite family of $j$-invariant zero elliptic curves $E_{D}:y^{2}=x^{3}+16D$ and their $λ$-isogenous curves $E_{D'}:y^{2}=x^{3}-27\cdot16D$, where $D$ and $D' = -3D$ are fundamental discriminants of a specific form, and $λ$ is an isogeny of degree $3$. A result of Honda guarantees that for our discriminants $D$, the quadratic number field $K_{D} = \mathbb{Q}(\sqrt{D})$ always has non-trivial 3-class group. We prove a series of results related to the set of rational points $E_{D'}(\mathbb{Q}) \setminus λ(E_{D}(\mathbb{Q}))$, and the $SL(2,\mathbb{Z})$-equivalence classes of irreducible integral binary cubic forms of discriminant $D$. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of $E_{D}$ and the rank of its $3$-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-$1$ curves that correspond to irreducible integral binary cubic forms of discriminant $D=48035713$, and we show that every curve in these classes violates the Hasse Principle.

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

Abstract

We study an infinite family of -invariant zero elliptic curves and their -isogenous curves , where and are fundamental discriminants of a specific form, and is an isogeny of degree . A result of Honda guarantees that for our discriminants , the quadratic number field always has non-trivial 3-class group. We prove a series of results related to the set of rational points , and the -equivalence classes of irreducible integral binary cubic forms of discriminant . By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of and the rank of its -Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus- curves that correspond to irreducible integral binary cubic forms of discriminant , and we show that every curve in these classes violates the Hasse Principle.

Paper Structure

This paper contains 19 sections, 21 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Unramified cubic extensions of quadratic number fields
  3. The $3$-part of the ideal class group.
  4. Cubic fields of fundamental discriminant.
  5. Our Discriminants $D$
  6. Unramified cubic extensions and $3$-virtual units
  7. The elliptic curves $E_{D}$
  8. Descent by $3$-isogeny and the Selmer groups of $E_{D}$ and $E_{D'}$.
  9. The Fundamental $3$-Descent Map
  10. Binary Cubic Forms: Known Facts and Definitions
  11. The group of rational points $E_{D'}(\mathbb{Q})$.
  12. From IBCF's to Index Forms
  13. On the rank of $E_{D}/\mathbb{Q}$
  14. On the parity of the rank
  15. Lower bounds
  16. ...and 4 more sections

Key Result

Proposition 1.1

For every $D \in \mathcal{D}$, the rank of the Selmer groups $\mathcal{S}_{\lambda}$ and $\mathcal{S}_{\lambda'}$ of the curves $E_{D}$ and $E_{D'}$ are as follows:

Figures (1)

  • Figure :

Theorems & Definitions (45)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 35 more