Questions tagged [local-fields]
The local-fields tag has no usage guidance.
262 questions
4
votes
0
answers
190
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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
2
votes
0
answers
111
views
split tori over local fields
Let $F$ be a non-archimedean local field, and $\mathscr O$ its ring of integers. Suppose $T$ is an $F$-split torus, i.e., $T = (\mathbb G_m)^r$ where $\mathbb G_m$ denotes the multiplicative group. ...
- 303
2
votes
1
answer
167
views
Weil group of a local field, small notational problem
In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following:
Let $F$ be a non-archimedean local field, $\mathcal W_F$ its ...
- 303
8
votes
2
answers
725
views
Is every connected reductive group over a local field already defined over a global field?
Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...
- 113
1
vote
1
answer
228
views
Structure of locally compact non discrete topological division algebras without the use of Haar measure
There is a well-known structure theorem for locally compact non discrete topological division algebras, see here
https://math.stackexchange.com/q/1160086/187521
(I repost it here because I think it ...
- 255
4
votes
0
answers
322
views
Automorphisms of k((X))
I'm looking for a good reference for the following fact:
Let $k$ be a perfect field of characteristic $p$ and let $K=k((X))$.
Then every $k$-linear automorphism of $K$ is continuous with respect
to ...
19
votes
1
answer
4k
views
A good book on adeles and ideles
Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...
- 5,424
3
votes
1
answer
245
views
Is $G \rightarrow G/P$ surjective on $K$-points over a local field?
Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
- 1,103
11
votes
1
answer
2k
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On unramified p-adic groups
Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit ...
- 8,866
2
votes
2
answers
552
views
Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact ...
- 1,103
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
3
votes
1
answer
492
views
Theorem 7b of Serre's "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques"
Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...
- 31
7
votes
2
answers
350
views
explicit uniformizer for the false Tate extension
Let $p$ be an odd prime and let $n\geq 1$. Set $K=\mathbb{Q}_p(\zeta_{p^n})$,
$L=\mathbb{Q}_p(\sqrt[p^n]{p})$, and $M=KL$. I claim that $M$ is totally ramified of degree $\phi(p^n)p^n$ (the proof ...
- 7,609
6
votes
3
answers
1k
views
Finite extension of local fields
Can a (higher) local field have uncountably many finite (seperable) extensions?
- 11.3k
1
vote
1
answer
173
views
Compact subgroups of linear groups over nonarchimedean fields
Let $n \in \mathbb{N}$, $K$ a (nonarchimedean) local field, $\overline{K}$ its algebraic closure. Take a compact subgroup $G \leq \text{GL}_n(\overline{K})$. Must there be a finite extension $F$ of $K$...
- 11.3k
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
3
votes
2
answers
828
views
Why is the norm map dual to restriction under Tate local duality?
Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
- 1,103
6
votes
0
answers
225
views
Parshin's buildings for higher local fields
What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over two-...
- 17.4k
22
votes
5
answers
2k
views
Local inverse Galois problem
It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedean field $K$ has solvable (in fact supersolvable [edit: no!]) Galois group $G$. One sees this by using the ramification ...
- 1,062
1
vote
1
answer
324
views
Maximal separable extension of $\mathbb F_q((t))$
Let $K=\mathbb F_q((t))$. I want to prove that $K^{sep}$ is composite of $K^{sep}(p)$ and $K^{sep}(not \ p)$, where $K^{sep}(p)$ is maximal Galois extension of $K$ of exponent $p$, $K^{sep}(not \ p)$ ...
- 11
2
votes
0
answers
433
views
Global Artin reciprocity law from Local class field theory
Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...
- 71
5
votes
1
answer
516
views
Reference for Local class field theory via witt vectors
I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...
- 71
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
9
votes
1
answer
448
views
Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields
I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a ...
- 2,011
6
votes
1
answer
598
views
Clarification about Tits' article in the Corvallis
I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ (...
- 899
0
votes
0
answers
129
views
Ramified complete discrete valuation rings as extensions
Suppose $O$ is a complete discrete valuation ring with uniformizer $\pi$ and residue field $k=O/\pi O$ of charactersitc $p>0$. If $\nu$ is the $\pi$-adic valuation on $K=Frac(O)$, suppose also ...
- 298
5
votes
1
answer
445
views
Existence of maximal totally ramified $p$-extension of a local field
This relates to this question:
Existence of maximal totally ramified extensions of an arbitrary CDVF
Let $K$ be a local field with finite residue field of characteristic $p>0$. Does there exist a ...
- 1,661
2
votes
0
answers
415
views
Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field
Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...
- 1,457
7
votes
0
answers
284
views
Epsilon factors for tamely ramified extensions of local fields
Let $F$ be an unramified extension of $\mathbf{Q}_p$ of degree $n$, and let $K = F(\alpha)$ where $\alpha$ satisfies $\alpha^{p^n - 1} = -p$.
I'm interested in the local $\varepsilon$-factors ...
- 37k
0
votes
1
answer
346
views
a question about a result in Bushnell-Henniart book 'the local Langlands conjecture for GL(2)'
This might be a easy question, but I couldn't get the point.
Let $F$ be a p-adic field, $\bar{F}$ a separable algebraic closure of $F$. Set $\Omega_F=Gal(\bar{F}/F)$. Use $F_{\infty}\subset \bar{F}$ ...
- 2,709
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 $\mathbb Q_p,...
- 51
4
votes
1
answer
202
views
Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$
Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over $\...
- 5,019
5
votes
1
answer
337
views
Is $SL_1(D)$ toplogically finitely generated, for $D$ a division algebra over a local field?
I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...
- 993
4
votes
1
answer
469
views
Semisimple group not split by an unramified extension?
Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?
- 1,704
10
votes
4
answers
2k
views
Reference request: expository text on the structure of reductive groups over non-archimedean local fields
I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner (...
- 2,027
8
votes
0
answers
221
views
Inertia group vs. differential equations
The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...
- 2,040
2
votes
3
answers
583
views
A question on Haar measure on local field.
Let $F$ be a local field of characteristic 0, and $f:F\rightarrow \mathbb{C}$ be an integrable function. Is the following formulation valid?
$
\int_{F^\times}f(x^2) d^\times x=\int_{F^{\times 2}}f(x) ...
- 833
9
votes
1
answer
4k
views
Maximal tamely ramified extension of $\mathbf Q_p$
Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?
- 2,040
11
votes
1
answer
4k
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henselization and completion
This might not be a question appropriate for this forum, I apologize in this case...
Is it true that any element of the completion of a valued ring $R$ that is algebraic over the field of fractions of ...
- 111
3
votes
1
answer
377
views
What's the minimum number of generators for the wild inertia?
Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be ...
- 5,019
11
votes
0
answers
383
views
Galois invariants in a ring of fractional power series over a finite field
Let $\mathbf{F}_q$ be a finite field, and let $A=\mathbf{F}_q [[ x^{1/q^\infty} ]]$ be the completion of $\mathbf{F}_q[x^{1/q^\infty}]$ with respect to the $x$-adic topology. Then the $q$th power ...
- 4,487
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
1
vote
1
answer
271
views
Local densities of hermitian forms
I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see 1).
I would have liked to create the ...
- 2,446
4
votes
1
answer
751
views
How to understand the representation theory of $SL(n)$ from $GL(n)$?
Let $F$ be a local field. Consider the group extension (split)
$$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$
What knowledge about $PGL(n)$ is necessary in order to understand ...
- 11.2k
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
2
votes
3
answers
889
views
squares in dyadic local fields
Hello,
By the local square theorem I know that $1+4\alpha$ is square if $|\alpha|<1$ ($\alpha$ is not a unit). Now, Can I always get a unit $\alpha$ such that $1+4\alpha$ is not a square ?? For ...
- 171
1
vote
3
answers
873
views
Brauer group of complete DVR
Let $A$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $\kappa$.
Let $K_{nr}$ be the maximal unramified extension of $K$ and let $A_{nr}$ be its ring of ...
- 5,163
9
votes
2
answers
626
views
Invariant functor for admissible representations of reductive groups over local fields
Hello,
I have a question concerning a certain functor between represention categories. I'm rather sure this is already known, but I could not find a reference.
Let $F$ be a local non-archimedean ...
- 223
4
votes
4
answers
836
views
cuspidal types and Iwahori subgroup for $SL(2,F)$
Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.
Is there any possibility that $J\subset I$ or even a subgroup?
- 85
3
votes
1
answer
871
views
Discrete Series representations for $SL_{2}$ over $p$-adic field.
I am working on the chamber homology for $SL(2,F)$, and stuck at some basic stuff on D.S. reps of $SL(2,F)$.
Let $ I=\left(
\begin{array}{cc}
\mathcal{O}_{F} & \mathcal{O}_{F} \\
...