18
votes
Accepted
Is the Tate-Shafarevich group of a rational elliptic curve finite?
MO is not the place to discuss the validity of preprints, but I think it is safe to say that the finitiness of the Tate-Shafarevich group for elliptic curves over $\mathbb{Q}$ is considered an open ...
- 17.6k
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 ...
- 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 ...
- 8,945
6
votes
Accepted
The second Tate-Shafarevich group of a permutation module is trivial
We write $G_w={\rm Gal}(L_w/K_v)$.
Definition. For $n\ge 1$, we denote
$$Ш_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod_C H^n(C,M)\Big)$$
where $C$ runs over the cyclic subgroups of $G$.
Remark. $Ш^2(L/...
- 14.2k
3
votes
Accepted
Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
This follows from Theorem 1.5 of Alex Smith's paper "The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I" which states
Suppose $A/\mathbb Q$ is an elliptic ...
- 149k
3
votes
Accepted
Local triviality of Galois cohomology classes over $\mathbb{Q}$
I think it follows again from Chebotarev. Represent the cohomogy class as an extension of the trivial representation by $A$. Such an extension is itself a Galois action on a finitely-generated $\...
- 149k
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