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Jul
31
comment What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?
The group law, which I do believe you're asking for (as part of your question), is relatively simple. Namely, since $E/k$ is a group scheme, you have a multiplication map $\mu:E\times_k E\to E$. This maps $(e,e)$ to $e$ (the identity element) and so gives you a map $\mathcal{O}_{E,e}\to \mathcal{O}_{E\times E,(e,e)}$. Passing to the completions, and using the fact that both are smooth, we get a map $k[[T]]\to k[[X,Y]]$. The image of $T$ is the formal group law associated to $E$.
Jul
27
awarded  Enlightened
Jul
27
awarded  Nice Answer
Jul
23
comment Difference between $K$-rational points and $K$-valued points
Containment of rings=field extension for fields. The terms are synonomous. And so yes, K-rational and K-points mean the same thing. :)
Jul
23
comment Neukirch's motivation for $p$-adic numbers
@Lubin I disagree in some sense. I think the analogy I mentioned above is pretty concrete. There is also this issue with regards to power series (albeit easier to fix). Abstractly $\widehat{\mathcal{O}_{X,x}}$ is a power series ring, and choosing an explicit isomorphism is choosing a center for the power series expansion. So, there are no unique coefficients, but 'natural ones', similarly to the case of p-adics.
Jul
22
comment Self-intersection number of fibered surface
I don't have it on hand, but if it's somewhere accessible, it'll be in chapter 9 of Qing Liu.
Jul
22
comment Weil: Fibre Spaces in Algebraic Geometry
Why don't you request it from a library? A few libraries near me have it. worldcat.org/title/fibre-spaces-in-algebraic-geometry/oclc/…
Jul
21
comment Continuity of Galois representations from cohomology
@dionysos It is for the finite pieces--the Z/l^nZ coeffs.
Jul
21
comment What is the degree of the pull back of a line bundle?
What is your definition of degree? Some people may define degree as $\deg(i^\ast\mathcal{O}(1))$.
Jul
21
comment Continuity of Galois representations from cohomology
$\mathrm{Hom}_{\mathrm{Spec}(\mathbb{Q})}(-,\mathrm{Spec}(L))$ for some number field $L$. The kernel is then $\mathrm{Gal}(\overline{\mathbb{Q}}/L)$. I'm not sure how to state this intrinsically in this generality. For elliptic curves it's simple since $H^1$ is dual to $T_\ell$ which clearly is continuous, and the higher $H^i$ are just wedges. I don't know something as concrete as 'adjoin torsion points to $\mathbb{Q}$', like for ab. vars, in the generality you ask. Let me know if you think of any neat way of characterizing $L$.
Jul
21
comment Continuity of Galois representations from cohomology
@dionysos No problem! Glad I could help. I'll have to think about the easiest way to state continuity without thinking this way. You see, the problem is, as you pointed out, it's enough to show continuity of the action on $H^i(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})$. This amounts to saying that $G_\mathbb{Q}\to \mathrm{GL}_{\mathbb{Z}/\ell^n\mathbb{Z}}(H^i(\overline{X},\mathbb{Z}/\ell^n \mathbb{Z}))$ factors through a finite quotient. This quotient is not obvious to me except to say that $R^if_\ast\underline{\mathbb{Z}/\ell^n\mathbb{Z}}$ is of the form
Jul
21
comment Factoring ideals in algebraic number rings using Dedekind's theorem
@JyrkiLahtonen No problem :) I fear that this is precisely the answer the OP didn't want though. So, we'll see.
Jul
21
comment Factoring ideals in algebraic number rings using Dedekind's theorem
@JyrkiLahtonen He has containment of ideals in one direction, and I just showed that their quotients are isomorphic (so have the same cardinality) which implies that the ideals are actually equal.
Jul
21
comment Serge Lang Never Explains Anything Round II
Don't you just put any two abelian extension in a third, and then use the fact that your two supposed different elements of $G$ are the restriction of the same element upstairs?
Jul
21
answered Factoring ideals in algebraic number rings using Dedekind's theorem
Jul
21
revised Continuity of Galois representations from cohomology
edited body
Jul
21
comment Something about closed sets in $\mathbb{P}^n\times\mathbb{P}^m$
What do you mean 'generated by'?
Jul
21
answered Continuity of Galois representations from cohomology
Jul
21
comment $R^kp_{2,*} (p_1^* V\otimes P) =0$ for $k\neq g$ and $V$ is $\pi_*$ acyclic on abelian schemes?
Can you give some motivation for this problem? It'd be more interesting to work on if we knew why we care.
Jul
21
comment Neukirch's motivation for $p$-adic numbers
derivatives. One can see this if you think of $\widehat{\mathcal{O}_{X,p}}$ as $\varprojlim \mathcal{O}_{X,p}/\mathfrak{m}_p^n$. Mapping $f$ in $\mathcal{O}_X(X)$ to $\mathcal{O}_{X,p}/\mathfrak{m}_p$ is like taking the constant term, the ring $\mathcal{O}_{X,p}/\mathfrak{m}_p^2$ is like taking the linear term, etc. So, the inverse limit is like taking the Taylor expansion.