All Questions
Tagged with local-fields ag.algebraic-geometry
14 questions with no upvoted or accepted answers
10
votes
0
answers
409
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Higher Adeles of a scheme and related topics
Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar).
Consider the augmented ...
- 7,799
6
votes
0
answers
370
views
What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?
Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
- 3,625
5
votes
0
answers
299
views
A relative version of Hensel's lemma?
Let $k$ be a $p$-adic field with integer ring $\mathcal{O}_k \subseteq k$, maximal ideal $m_k \subseteq \mathcal{O}_k$ and residue field $\mathbb{F}_q = \mathcal{O}_k/m_k$. Let $X$ be a smooth, ...
- 9,491
4
votes
0
answers
228
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The definition of complex multiplication on K3 surfaces
I am reading this paper on the complex multiplication of K3 surfaces. It seems that this is only defined for complex K3 surfaces, or K3 surfaces over number fields. Is there a more general defintion ...
- 141
4
votes
0
answers
157
views
Reference request - conjugacy classes over local fields
Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in ...
- 5,562
4
votes
0
answers
1k
views
Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
- 11.2k
3
votes
0
answers
135
views
Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group
There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1):
$S$ is a Henselian trait (i.e. the spectrum of a henselian discrete ...
- 83
2
votes
0
answers
74
views
Arbitrary base change of a parahoric subgroup in split case
Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...
- 291
2
votes
0
answers
258
views
Is a reductive group scheme always parahoric?
Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
- 407
2
votes
0
answers
115
views
Converging sequence of base change
Here is a natural question that I hope will be of interest to some.
Let $\mathbf{F}_p(\!(T)\!)$ be the field of formal Laurent series over $\mathbf{F}_p$. An automorphism of $\mathbf{F}_p(\!(T)\!)$ ...
- 1,017
2
votes
0
answers
72
views
Continuity of the solutions of an isogeny in a formal group
Notation for the problem:
Let $E/\mathbb{Q}_P$ be a local field, and $\mu_E$ its maximal ideal. Let $K=E\{\{T\}\}$ be the standard 2-dimensional local field equipped with the Parshin topology and let ...
- 21
1
vote
0
answers
105
views
Algebraic morphisms of affine varieties in positive characteristic
Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$.
Consider two matrices $M_1,M_2$ in $\mathcal M_2(\Omega)$ that are $\...
- 3,998
1
vote
0
answers
101
views
Relation between 1-dimensional and 2-dimensional reciprocity maps
Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, \...
- 33
0
votes
0
answers
175
views
Why $k((x,t))$ can not be a local field?
If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field.
I ...
- 930