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

- $r$ is the analytic rank of $A$;
- $L(A,s)$ is the L-function of $A$, with leading coefficient $L^{(r)}(A,1)/r!$;
- $A(\Q)_{\rm tor}$ is the torsion subgroup of the Mordell-Weil group $A(\Q)$;
- $\Omega$ is the real period of $A$;
- $R$ is the regulator of $A$;
- $T = \prod_p c_p$ is the Tamagawa product of $A$.

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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-09 17:10:48

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**History:**(expand/hide all)

- 2020-10-09 17:10:48 by Andrew Sutherland (Reviewed)
- 2020-10-09 17:09:56 by Andrew Sutherland
- 2020-01-04 07:09:40 by Andrew Sutherland (Reviewed)
- 2020-01-04 07:03:45 by Andrew Sutherland
- 2020-01-03 22:27:24 by Andrew Sutherland

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