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Selmer ranks under quadratic twists satisfying the Heegner hypothesis

Alexandros Konstantinou

TL;DR

The article analyzes how 2-Selmer ranks of an elliptic curve with partial 2-torsion change under quadratic twists that satisfy the Heegner hypothesis. It introduces a common relaxed Selmer framework and encodes local-twist effects as linear-algebra over the field $\mathbb{F}_{2}$, yielding explicit rank-change formulas in terms of matrices built from Legendre symbols. The main contributions are the detailed expressions for $\delta_{\varphi}(d)$ and $\delta_{\hat{\varphi}}(d)$ across sign configurations, and the demonstration that these changes align with the parity predictions of the $2$-parity conjecture, including a symmetric relation for Tate–Shafarevich groups. The results provide computable, parity-consistent predictions for Selmer ranks in Heegner twist families, with broader implications for the arithmetic of quadratic twists and parity phenomena. Overall, the work connects explicit local data and global Selmer structure to confirm parity expectations in a rich Heegner-twist setting.

Abstract

We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve $E/\mathbb{Q}$ with partial $2$-torsion and a common relaxed Selmer group, we derive explicit formulae describing the effect of twisting on Selmer ranks in terms of matrices over $\mathbb{F}_{2}$. As an application, we show that these formulae are compatible with predictions made by the parity conjecture.

Selmer ranks under quadratic twists satisfying the Heegner hypothesis

TL;DR

The article analyzes how 2-Selmer ranks of an elliptic curve with partial 2-torsion change under quadratic twists that satisfy the Heegner hypothesis. It introduces a common relaxed Selmer framework and encodes local-twist effects as linear-algebra over the field , yielding explicit rank-change formulas in terms of matrices built from Legendre symbols. The main contributions are the detailed expressions for and across sign configurations, and the demonstration that these changes align with the parity predictions of the -parity conjecture, including a symmetric relation for Tate–Shafarevich groups. The results provide computable, parity-consistent predictions for Selmer ranks in Heegner twist families, with broader implications for the arithmetic of quadratic twists and parity phenomena. Overall, the work connects explicit local data and global Selmer structure to confirm parity expectations in a rich Heegner-twist setting.

Abstract

We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve with partial -torsion and a common relaxed Selmer group, we derive explicit formulae describing the effect of twisting on Selmer ranks in terms of matrices over . As an application, we show that these formulae are compatible with predictions made by the parity conjecture.
Paper Structure (22 sections, 27 theorems, 102 equations)

This paper contains 22 sections, 27 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Elliptic curves over quadratic extensions
  3. Effect of twists satisfying the Heegner hypothesis
  4. Structure of the argument
  5. Parity conjecture and twisting
  6. General Notation
  7. Selmer groups and local conditions
  8. Selmer groups and local subgroups
  9. Variation of $\varphi$–Selmer groups under quadratic twisting
  10. Twisting by imaginary quadratic fields satisfying the Heegner hypothesis
  11. Twisting local conditions when $v | d$
  12. Twisting local conditions when $v = \infty$
  13. Truncated structures and modified Selmer groups
  14. Truncated and sign–modified Selmer structures
  15. Linear–algebraic description of Selmer conditions
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $E/\mathbb{Q}$ be given by the equation $y^{2}=x(x^{2}+a x+b)$, and let $d\in\mathcal{D}_{E/\mathbb{Q}}$. Define The values of $\delta_{\varphi}(d)$ and $\delta_{\hat{\varphi}}(d)$ are given in the following table. Here, $\omega(d)$ denotes the number of prime factors of $d$, and all matrices $\mathbf{A}_{d}$, $\widetilde{\mathbf{A}_{d}}$, $\widehat{\mathbf{A}_{d}}$ and $\overline{\mathbf{A}

Theorems & Definitions (63)

  • Theorem 1.1: = Thm. \ref{['thm:master2']}, Cor. \ref{['cor:hatphi-variation-explicit']}
  • Conjecture 1.2: Parity conjecture
  • Corollary 1.6: = Corollary \ref{['cor:symmetric-sha-ratio']}
  • Lemma 2.3: Cassels1965
  • proof
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 53 more