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comment 
Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?
What is $p$ and $n$ ? It seems to me that the linked article calls it the "stong discrete logarithm" and it discusses the issue. (I have not read beyond the first page). What exactly is your question ? 
Jun 29 
comment 
Fermat's Last Theorem in $\mathbb{Z}/n\mathbb{Z}$
And an answer to your question is $(2,2,1)$. In summary, this question does not sound like a research level question. 
Jun 29 
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Fermat's Last Theorem in $\mathbb{Z}/n\mathbb{Z}$
For $(x,y,z)\in\mathbb{N}^3$. If there is a prime $n$ dividing all or just one of the three then $(x,y,z)\in R$. (Take $k=n1$ for the second case.) 
Jun 25 
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Conditions for splitting of short exact sequence?
It is just a short exact sequence of $m$torsion abelian groups. It quite easy to give examples and conditions for both cases. I don't think you can hope for anything else. 
May 19 
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A Diophantine equation with prime powers
related to mathoverflow.net/questions/206931/… . 
May 18 
comment 
What is prime power of this equation of p?
@darya $19^2  19 + 1 = 7^3$ 
May 17 
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Reference request: Cohomology of Elliptic Curves
No. IF the $m$torsion points are defined over $K$, then all $Q\in E(\bar{K})$ with $mQ\in E(K)$ are defined over an abelian extension.  Instead the extension adjoining all $m$torsion points is a subgroup of $GL_2(\mathbb{Z}/m\mathbb{Z})$. It is very often equal to the full (very nonabelian) group. 
May 16 
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Reference request: Cohomology of Elliptic Curves
The group in question only makes sense if all poitns in $E_{p^n}$ are defined over $K^{ab}$. So you have to be in a rather particular situation unless they are already defined over $K$. 
May 5 
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n torsion groups of quadratic twists of elliptic curves
The representation $\rho^F$ of $G_K$ on $E^F[m]$ for any $m$ is the product of the representation $\rho$ on $E[m]$ times the (scalar) character $\chi$ corresponding to the extension $F/K$. If $m>2$ , there will be $\sigma$ in $G_K$ with $\chi(\sigma)=1$. Since $1$ and $+1$ are not congruent modulo $m>2$, we have $E^F[m]\neq E[m]$. 
Apr 6 
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Algorithm for fast factorization of polynomial over $\mathbb Z$ or over $\mathbb F_p$
Sage uses FLINT flintlib.org to do factorisation in $\mathbb{Z}[X]$ and $\mathbb{Z}/n\mathbb{Z}[X]$. For an introduction into the very basic factorisation algorithms, maybe Cohen's "A course in computational number theory" pp. 124 could help. 
Apr 6 
answered  Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties? 
Apr 6 
awarded  Fanatic 
Apr 4 
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Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties?
Sorry for the chain of comments, but the question is not (yet ?) open for an answer. 
Apr 4 
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Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties?
There is some evidence that the fine Sha is much smaller. For instance even over infinite extension considered in Iwasawa theory the fine Sha should be finite (all the time ?) while the full Sha can get very large. 
Apr 4 
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Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties?
As to the kernel of your map: This is what I would call the "fine TateShafarevich group". There are examples of when it is trivial, half or all of the TateShafarevich group when the latter has 4 elements. In general I would think the kernel could just be anything. Proc. Camb. Soc. 142 (2007), no. 1, p. 112. 
Apr 3 
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Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties?
(After edit): Take $\ell$primary parts everywhere. Then the local product of the first term is just $E(\mathbb{Q}_{\ell})\otimes\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}$ which is cofree of rank $1$. So $\operatorname{coker}(a)[\ell^{\infty}]$ is finite if the rank of $E(\mathbb{Q})$ is positive and is cofree of corank $1$ otherwise. That does not look analogous to your huge local product in $r$. 
Apr 1 
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Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties?
And the title of the question als odoes not seem to have much relation to the question itself. 
Apr 1 
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Is Ш a good parameter for the failure of GlobalLocal principle for abelian varieties?
What is your definition of $Sel(E/\mathbb{Q})$ ? My first guess would be the inductive limit of $n$Selmer groups. But then I can't see how you defined the map you want to be injective. Did you mean the target to be $E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z}$ ? I think this question needs some improvement to be understandable. 
Mar 30 
awarded  Yearling 
Mar 11 
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integral basis for the Lie algebra of the Neron model of an abelian variety
Lie$(A^{\vee})$ is defined to be the $O_K$dual of the differentials on the Néron model. So if that is not free, neither will Lie$(A^{\vee})$. But a basis of Lie${}_K(A^{\vee})$ is all that you need. 