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Results tagged with tate-shafarevich-groups
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user 5015
7
votes
Accepted
$S$-Tate-Shafarevich groups of elliptic curves
Global duality (as for instance on page 29 of Rubin's "Euler systems") gives you a description of the cokernel of your inclusion. Let $p$ be a prime. Then the $p$-primary part of the quotient of $Ш(E, …
- 8,945
2
votes
Is Ш a good parameter for the failure of Global-Local principle for abelian varieties?
All your groups are torsion, so we may split it into primary parts. Let $\ell$ be a prime. First the map $a\colon E(\mathbb{Q})\otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell} \to \prod_p E(\mathbb{Q}_p)\o …
- 8,945
8
votes
Accepted
Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$\DeclareMathOperator{\sha}{Ш}$
I am not sure that the proof that Sha has order 9 is anywhere spelled out in full. Here the ideas how to do it.
First, that the order of $C$ is three in the $\sha$ is j …
- 8,945
11
votes
Accepted
Relationship between Tate-Shafarevich group and the BSD conjecture
Firstly, the functional field result your state is due to Tate in his Bourbaki talk. In fact he proves that the finiteness of the $p$-primary part of Sha is enough for $p$ different from the character …
- 8,945
0
votes
Accepted
Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha...
Let $\sigma$ be the non-trivial element of the Galois group of the quadratic extension $L/K$. Let $\phi \colon E \to E_D$ be the isomorphism defined over $L$.
First, if $P \in E(\bar L)$ then $\sigma\ …
- 8,945
11
votes
Accepted
Discrepancy in Magma's calculation and Sage's of elliptic curve?
Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage.
In fact, when executed with higher values of the parameters so that the s …
- 8,945