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arXiv · 2411.15663

A Note on the Growth of Sha in Dihedral Extensions

Jamie Bell
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Abstract

We provide a formula for the order of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields of order $2n$, up to $4^{th}$ powers and primes dividing $n$. Specifically, for odd $n$ it is equal to the order of the Tate--Shafarevich group over the quadratic subextension. A similar formula holds for even $n$.

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Jamie Bell. 2024-11-23. A Note on the Growth of Sha in Dihedral Extensions. https://arxiv.org/abs/2411.15663

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