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For an abelian variety $A/\Q$ the analytic order of Sha is the real number \[ Ш_{an}(A) := \frac{L^{(r)}(A,1)}{r!}\frac{(\#A(\Q)_{\rm tor})^2}{\Omega_AR_AT_A} \] where

The Birch and Swinnerton-Dyer for abelian varieties implies that the analytic rank of $A$ is equal to its Mordell-Weil rank, and that its analytic order of Sha is equal to the order of its Tate-Shafarevich group.

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  • Last edited by Andrew Sutherland on 2020-01-04 07:09:40
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