# Questions tagged [tate-shafarevich-groups]

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### 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 ...
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### 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 $v$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...
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### 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 ...
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### 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)...
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, ...