Proves that ratios of Sha[4] over quadratic extensions and Sha[2] of twists can grow arbitrarily large without assuming finiteness, and that Sha[2] is bounded with vanishing twists for infinitely many D in the family y^2 = x^3 + p x assuming finiteness.
Cassels, Arithmetic on curves of genus 1
2 Pith papers cite this work. Polarity classification is still indexing.
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Visibility theorems imply nontrivial ℓ-torsion in Sha of quadratic twists of elliptic curves with additive reduction at ℓ; for ℓ=3 this yields pairs of curves with identical BSD data and Kodaira symbols but isomorphic Sha groups containing 3-torsion.
citing papers explorer
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Behaviors of the Tate--Shafarevich group of elliptic curves under quadratic field extensions
Proves that ratios of Sha[4] over quadratic extensions and Sha[2] of twists can grow arbitrarily large without assuming finiteness, and that Sha[2] is bounded with vanishing twists for infinitely many D in the family y^2 = x^3 + p x assuming finiteness.
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Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists
Visibility theorems imply nontrivial ℓ-torsion in Sha of quadratic twists of elliptic curves with additive reduction at ℓ; for ℓ=3 this yields pairs of curves with identical BSD data and Kodaira symbols but isomorphic Sha groups containing 3-torsion.