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.

**Knowl status:**

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

**Referred to by:**

- ec.bsdconjecture
- ec.q.analytic_sha_value
- ec.q.bsd_invariants
- ec.q.bsdconjecture
- rcs.rigor.ec
- rcs.rigor.ec.q
- lmfdb/ecnf/main.py (line 390)
- lmfdb/ecnf/main.py (line 804)
- lmfdb/ecnf/templates/ecnf-curve.html (lines 318-320)
- lmfdb/elliptic_curves/elliptic_curve.py (line 197)
- lmfdb/elliptic_curves/elliptic_curve.py (lines 466-467)
- lmfdb/elliptic_curves/elliptic_curve.py (line 1233)
- lmfdb/elliptic_curves/elliptic_curve.py (line 1267)
- lmfdb/elliptic_curves/templates/congruent_number_data.html (line 175)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 279)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 299)
- lmfdb/elliptic_curves/templates/ec-stats.html (line 139)

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

- 2024-03-08 03:42:47 by John Cremona
- 2024-03-07 09:52:40 by John Cremona
- 2023-11-15 06:46:45 by John Cremona
- 2023-11-03 12:21:02 by John Cremona
- 2023-11-03 11:04:11 by John Cremona
- 2023-11-03 10:56:51 by John Cremona
- 2023-11-02 12:43:51 by John Cremona
- 2020-10-13 18:14:47 by Andrew Sutherland (Reviewed)
- 2020-10-13 16:04:37 by Andrew Sutherland
- 2020-10-12 16:22:47 by Kiran S. Kedlaya
- 2020-10-09 12:35:25 by John Cremona
- 2020-10-08 04:45:37 by John Cremona
- 2020-10-08 04:26:17 by John Cremona

**Differences**(show/hide)