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Nontrivial torsion in the Tate--Shafarevich group via visibility and twists

Asuka Shiga

TL;DR

The paper develops a visibility framework for elliptic curves over $\mathbb{Q}$ with additive reduction at an odd prime $\ell$ and no $\mathbb{Q}$-rational degree $\ell$-isogeny, establishing a twisted-visibility theorem that yields nontrivial $\ell$-torsion in $\Sha(E^D/\mathbb{Q})$ for suitable square-free twists $D$. It introduces a relaxed $\ell$-torsion subgroup $\Sha(A/\mathbb{Q},M)[\ell]$ and analyzes visibility via quotients and local conditions, including the handling of bad reduction primes. As an explicit application with $\ell=3$, the paper shows that non-isomorphic pairs of elliptic curves can share identical BSD data and Kodaira symbols while possessing nontrivial $3$-primary parts of their Tate–Shafarevich groups, demonstrated using pairs like $E_1=38025.ck1$ and $F=38025.i1$ with twists $D=6977$ and $D=23297$, yielding $\Sha(E_1^D/\mathbb{Q})[3]\neq 0$ and $\Sha(E_1^D/\mathbb{Q})[3]\cong\Sha(E_2^D/\mathbb{Q})[3]$. These results enrich the landscape of $3$-torsion phenomena in $\Sha$ and illustrate how visibility combined with twists can produce new, data-stable examples beyond existing tables.

Abstract

Let $\ell$ be an odd prime. We study the visibility theorem for certain elliptic curves over $\mathbb{Q}$ with additive reduction at $\ell$ and no degree $\ell$ isogeny defined over $\mathbb{Q}$, and deduce the existence of nontrivial $\ell$-torsion in $\Sha(E^D/\mathbb{Q})$ for suitable quadratic twists $E^D$. As an application for $\ell=3$, we exhibit pairs of non-isomorphic elliptic curves with the same BSD invariants, Kodaira symbols, and minimal discriminants, whose Tate--Shafarevich groups are isomorphic and have nontrivial $3$-primary parts.

Nontrivial torsion in the Tate--Shafarevich group via visibility and twists

TL;DR

The paper develops a visibility framework for elliptic curves over with additive reduction at an odd prime and no -rational degree -isogeny, establishing a twisted-visibility theorem that yields nontrivial -torsion in for suitable square-free twists . It introduces a relaxed -torsion subgroup and analyzes visibility via quotients and local conditions, including the handling of bad reduction primes. As an explicit application with , the paper shows that non-isomorphic pairs of elliptic curves can share identical BSD data and Kodaira symbols while possessing nontrivial -primary parts of their Tate–Shafarevich groups, demonstrated using pairs like and with twists and , yielding and . These results enrich the landscape of -torsion phenomena in and illustrate how visibility combined with twists can produce new, data-stable examples beyond existing tables.

Abstract

Let be an odd prime. We study the visibility theorem for certain elliptic curves over with additive reduction at and no degree isogeny defined over , and deduce the existence of nontrivial -torsion in for suitable quadratic twists . As an application for , we exhibit pairs of non-isomorphic elliptic curves with the same BSD invariants, Kodaira symbols, and minimal discriminants, whose Tate--Shafarevich groups are isomorphic and have nontrivial -primary parts.
Paper Structure (11 sections, 15 theorems, 58 equations, 3 figures, 2 tables)

This paper contains 11 sections, 15 theorems, 58 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and outline of the proof
  3. Notation
  4. Structure of the paper
  5. $\ell$-congruent elliptic curves
  6. Visibility
  7. Theory of visibility
  8. The relaxed version of $\ell$-torsion subgroup of the Tate--Shafarevich group
  9. Visibility for twisted elliptic curves at bad reduction
  10. Non-isomorphic pairs of elliptic curves with identical BSD invariants, Kodaira symbols, minimal discriminants, and nontrivial Tate–Shafarevich groups
  11. Acknowledgements

Key Result

Theorem 1.1

(=Theorem main, A visibility theorem for twisted elliptic curves, including primes of bad reduction) Let $E,F$ be elliptic curves over $\Bbb{Q}$. Fix an odd prime number $\ell$. Assume that $(E,F)$ satisfies the following conditions: Then,

Figures (3)

  • Figure 1: If $\#(B^D(\Bbb{Q})/\ell B^D(\Bbb{Q}))\ge \ell^2$ and $\#\Sha(A^{D}/\mathbb{Q}, \text{Vis})[\ell]/\#\Sha(A^{D}/\mathbb{Q})[\ell] < \ell^{2}$, then $\Sha(A^{D}/\mathbb{Q})[\ell]\neq 0$.
  • Figure :
  • Figure :

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1: Cor 9.19 of stein, Sturm bound
  • Theorem 3.2: Cor 9.19 of stein
  • Example 3.3
  • Theorem 3.4: Brauer--Nesbitt, see Theorem 2.9 of Wiese2013
  • Theorem 3.5
  • proof
  • Definition 3.6
  • Example 3.7: $\ell=3$
  • ...and 28 more