# Questions tagged [tate-shafarevich-groups]

The tate-shafarevich-groups tag has no usage guidance.

16
questions

0
votes

0
answers

75
views

### Reference of R. Kloosterman

Here, Clark suggests to use the theorem of R. Kloosterman, and then mentions one of his papers. I searched this theorem in his paper but I didn't find anything close to this theorem. Where can I find ...

2
votes

0
answers

176
views

### Relation between the Tate-Shafarevich group of a number field and the Tate-Shafarevich group of an elliptic curve

Let $E$ be an elliptic curve over a number field $K$, and let $sha^1(K,E)$ be the Tate-Shafarevich group, defined by:
Let $v$ be a valuation on $K$, and denote by $K_v$ the completion of $K$ by $v$, ...

2
votes

0
answers

131
views

### Understanding Sha through $K_2$

Consider the following setup, to keep things easy: let $F$ be a number field with ring of integers $A$. Let $E$ be an elliptic curve over $F$ with Neron model $N$ over $A$. Let $Sha(E)$ be the ...

3
votes

0
answers

110
views

### Some Questions regarding the fine Shafarevich-Tate group

The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes ...

1
vote

0
answers

182
views

### Growth of Selmer Groups

If $E$ is an elliptic curve over $K$, is there any effective estimate for the discriminant of the extension $L/K$ for which the $p$-part of the
Selmer or Tate-Shafarevich groups become large?
I will ...

7
votes

0
answers

142
views

### Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$ and certain other properties

I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...

3
votes

1
answer

1k
views

### Is the Tate-Shafarevich group of a rational elliptic curve finite?

It seems that Lan Nguyen proved in a preprint on arxiv of 2013 that the Tate-Shafarevich group of a rational elliptic curve is finite. However, I couldn't find any published version thereof. So is it ...

4
votes

1
answer

236
views

### Local triviality of Galois cohomology classes over $\mathbb{Q}$

Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in
$$\mathrm{H}^1(\mathbb{...

6
votes

1
answer

737
views

### Relationship between Tate-Shafarevich group and the BSD conjecture

The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken....

3
votes

1
answer

370
views

### Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many implications of the Birch and Swinnerton-Dyer ...

2
votes

1
answer

356
views

### Is Ш a good parameter for the failure of Global-Local principle for abelian varieties?

(Comparing to class group cases: we have an isomorphism
$Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.
Similarly, for an elliptic curve $E/\...

7
votes

1
answer

435
views

### Tate-Shafarevich groups over finitely generated fields

Let $G$ be an algebraic group over a number field $k$. One defines the Tate-Shafarevich set of $G$ to be
$$Ш(k,G) = \ker\left(H^1(k,G) \to \prod_{v} H^1(k_v,G)\right),$$
where the product is over all ...

13
votes

1
answer

635
views

### $S$-Tate-Shafarevich groups of elliptic curves

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be
$$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v \...

73
votes

0
answers

3k
views

### The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p)...

-2
votes

1
answer

1k
views

### Why should I believe in the Siegel's and Hasse's rationale ?

Hello everyone,
I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, ...

17
votes

1
answer

1k
views

### What's the Hilbert class field of an elliptic curve?

My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first.
Let E be an elliptic curve defined over some ...