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Explicit 7-torsion in the Tate–Shafarevich groups of genus 2 Jacobians

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An algorithm reveals non-trivial order 7 elements in Tate–Shafarevich groups of genus 2 jacobians, indicating unique elliptic curve relationships.

Non-trivial elements of order 7 exist in the Tate–Shafarevich groups of some genus 2 jacobians, indicating unique properties.
The algorithm generates twists of the Klein quartic curve related to genus 2 jacobians, revealing remarkable Galois representations.
Genus 2 curves with small conductor from Bending and Elkies–Kumar families demonstrate interesting order 7 torsion.
An abelian three-fold reveals the visibility of these non-trivial order 7 group elements, enhancing understanding of elliptic curves.

Sam Frengley (2025) studied this question.

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View Full Paper

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An algorithm reveals non-trivial order 7 elements in Tate–Shafarevich groups of genus 2 jacobians, indicating unique elliptic curve relationships.

Non-trivial elements of order 7 exist in the Tate–Shafarevich groups of some genus 2 jacobians, indicating unique properties.
The algorithm generates twists of the Klein quartic curve related to genus 2 jacobians, revealing remarkable Galois representations.
Genus 2 curves with small conductor from Bending and Elkies–Kumar families demonstrate interesting order 7 torsion.
An abelian three-fold reveals the visibility of these non-trivial order 7 group elements, enhancing understanding of elliptic curves.

Sam Frengley (2025) studied this question.

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