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1d

comment 
Are there “adelic” Lfunctions?
What $L$functionwould you wnat to cook up with that. The charactersitic polynomial of Frobenius at $p$ is independent of which $\ell$ you take. So you recover the usual $L$function of the elliptic curve, no ? 
1d

comment 
Are there “adelic” Lfunctions?
What do you mean by an "adelic representation" ? Who acts on what ? 
May 19 
comment 
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 
comment 
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 
comment 
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 
comment 
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 
comment 
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 
comment 
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 
comment 
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 
comment 
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 
comment 
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 
comment 
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. 
Mar 10 
answered  integral basis for the Lie algebra of the Neron model of an abelian variety 
Mar 3 
comment 
If $E(K)=E(L)$ for an elliptic curve $E$ and an algebraic extension $L/K$, what can we say about $Sel(E/K), Sel(E/L), L/K$?
I assume that Sel$(E/K)$ is the group that fits into a short exact sequence between $E(K)$ and Sha$(E/K)$. Otherwise you need to fix your $n$ in $n$Selmer group (or your isogeny). So your question is equivalent to asking about the growth of the TateShafarevich group in the tower. Basically anything can happen. Iwasaw theory gives examples of exploding growth as well as frequent stabilisation. 