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The Tate-Shafarevich group Ш of an elliptic curve $E$ defined over a number field $K$ is a torsion abelian group, which can be defined in terms of Galois cohomology as $$Ш(E) := \ker\left(H^1(G_K,E) \to \prod_v H^1(G_{K_v},E_{K_v}) \right),$$ where $v$ runs over all places of $K$ (finite and infinite), $K_v$ is the completion of $K$ at $v$, $E_{K_v}$ is the base change of $E$ to $K_v$, and $G_K$ and $G_{K_v}$ denote absolute Galois groups.

The group Ш is conjectured to be finite, and its order appears in the strong form of the Birch-Swinnerton-Dyer Conjecture for $E$. The order implied by the conjecture is called the analytic order of Sha and can be defined as the real number $$Ш_{\text{an}} := \frac{\#E(K)_{\rm tor}^2 \cdot L^{(r)}(E,1) / r!}{|d_K|^{1/2}\cdot \Omega(E/K) \cdot \mathrm{Reg}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}}}.$$ It is known that if Ш is finite then its order is a square, so one expects the real number $Ш_{\text{an}}$ to always be a square integer.

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• Review status: reviewed
• Last edited by Andrew Sutherland on 2020-10-13 18:14:47
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