All Questions
16 questions with no upvoted or accepted answers
10
votes
0
answers
1k
views
Automorphisms of local fields
It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...
- 1,298
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
758
views
maximal abelian extension of quadratic extension of $\mathbb Q_p$
I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,...
- 51
4
votes
0
answers
190
views
Is $K^{ur} K^{\pi} = L$?
Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...
- 6,180
3
votes
0
answers
96
views
Multiplication law in a division algebra of dimension 9 over a non-archimedean local field
Let $k$ be a non-archimedean local field, for example, a $p$-adic field (a finite extension of the filed ${\Bbb Q}_p$ of $p$-adic numbers).
It is well known that there is a canonical isomorphism
$${\...
- 14.2k
3
votes
0
answers
347
views
Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves
I am trying to understand an argument of Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves. I am stuck and I would appreciate any explanations.
Let $E$ be an elliptic curve over $K$, ...
- 3,625
2
votes
0
answers
116
views
Restriction of the local Artin map on the valuation ring of a local field
Let $F$ be a local field, in particular a finite extension of $\mathbb{Q}_p$ for some prime $p$ and let $Art_{L}: L^\times \to Gal(F^{ab}/F)$ be its local Artin map. We know that if $L/F$ is a finite ...
- 367
2
votes
0
answers
338
views
Existence of "nth root function" which is analytic
Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...
- 123
2
votes
0
answers
729
views
On Serre's "Local fields"
While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to ...
- 445
2
votes
0
answers
181
views
Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
- 3,625
1
vote
0
answers
124
views
A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
- 424
1
vote
0
answers
255
views
Globalization of a local field
I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1.
Here is the statement.
...
user163485
1
vote
0
answers
212
views
Realization of a p-adic field as a completion of a number field
Let $F$ be a $p$-adic field of characteristic 0. Is it always possible to find a number field $K$ such that $K$ has only one place lying above $p$ and such that its completion at this place is $F$?
If ...
- 833
1
vote
0
answers
62
views
Valuations in Higher-dimensional local fields
I have the following question which I believ should be true but I would like to have a different opinion about it:
Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and $t_1,\...
- 11
0
votes
0
answers
93
views
Existence of maximal totally ramified subextension
Suppose $K/\mathbb Q_p$ is a finite extension, with ramification index $e$ and inertia index $f$. I want to ask whether there exists a subextension $L/\mathbb Q_p$ totally ramified of degree $e$?
- 785
0
votes
0
answers
78
views
Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$
Let $K$ be a local field of positive characteristic.
I'm looking for a $K$ which satisfies the following condition.
Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n)...
- 1,541