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3
votes
0answers
97 views

The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
3
votes
0answers
92 views

Metaplectic groups over non-archimedean local fields of characteristic>2

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the double cover metaplectic extension of symplectic groups $p: Mp_{2n}(K)\rightarrow ...
0
votes
0answers
68 views

Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
1
vote
0answers
45 views

Normgroup and the image of the Hilbert symbol are subgroups of index 2 in the principal units

Let $K$ be a local field over $\mathbb{Q}_2$ such that the extension $K(i)/K$ is ramified and let $U^1_{K(i)}$ and $U^1_K$ denote the groups of principal units in the fields $K(i)$ and $K$, ...
1
vote
0answers
38 views

Infinitesimally small elements in extensions of models of model-complete theories

Suppose that we have a first order language $\mathcal{L}$ that extends the language of rings. Let $T$ a be a topological $\mathcal{L}$-theory of fields in the sense of Pillay.. this means that not ...
2
votes
1answer
89 views

F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...
1
vote
1answer
59 views

Existence of the double coset ring on paper of Ihara

In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from ...
2
votes
2answers
117 views

Lubin-Tate modules and different uniformizers

Suppose I have a local field $\mathcal{O}_K$ and two different prime elements $\pi$ and $\overline{\pi},$ i.e they differ by a unit $\overline{\pi} = u \pi$ for some $u \in \mathcal{O}_K^{\times}$ not ...
2
votes
0answers
38 views

is there a criterion for a two-generator subgroup of $PL(2,K)$ to be a cocompact lattice?

In the case of the group $SO(n,1)$ there is a criterion known for whether or not two given elements of the group generate a cocompact lattice. Is any similar criterion known in the case of $PL(2,K)$ ...
6
votes
3answers
299 views

Argument of Zariski density to prove rationality of a regular map

Question: I want to know if the following result is correct: Let $k$ be a number field and $k_v$ be a completion of $k$ at some place $v$, denote $K_v$ an algebraic closure of $k_v$. ...
17
votes
2answers
697 views

Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...
5
votes
0answers
170 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, ...
5
votes
2answers
349 views

Finding the inertia group

Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$. What is the isomorphism class of the inertia group $I_p$, ...
1
vote
1answer
96 views

Measure of ramification of local fields using upper numbering

We let $F$ be a non-archimedean local field (say with finite residue field). Consider a Galois extensions $E$ of $F$, with $G = Gal(E/F)$, in a fixed separable closure $\bar{F}$ of $F$. Considering ...
3
votes
0answers
89 views

Can we prove the uniqueness of the local Artin map by using mostly global class field theory?

Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...
3
votes
0answers
143 views

Swan conductor, representation Weil group

Let $F$ be a non-archimedean local field and $\mathcal W_F$ its Weil group. We consider a linear representation $\sigma$ of $\mathcal W_F$. Could someone explain to me the definition of the Swan ...
2
votes
0answers
91 views

$K^{ur}K^{\pi} = L$

Let $K$ be a $p$-adic field, and $L$ an infinite abelian extension of $K$ containing $K^{ur}$. Let $\Phi: K^{\ast} \rightarrow Gal(L/K)$ be the local Artin map. Let $\pi$ be a uniformizer for $K$, ...
4
votes
0answers
130 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 $K$ ...
2
votes
0answers
72 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. ...
0
votes
1answer
98 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 ...
7
votes
2answers
332 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 ...
1
vote
1answer
95 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 http://math.stackexchange.com/q/1160086/187521 (I repost it here because I think it ...
4
votes
0answers
163 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 ...
6
votes
1answer
757 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, ...
3
votes
1answer
213 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?
7
votes
1answer
321 views

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 ...
2
votes
2answers
284 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
vote
0answers
84 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, ...
3
votes
1answer
284 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 $F'$ ...
4
votes
1answer
150 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 ...
6
votes
3answers
383 views

Finite extension of local fields

Can a (higher) local field have uncountably many finite (seperable) extensions?
1
vote
1answer
146 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 ...
2
votes
0answers
52 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 ...
2
votes
2answers
299 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 ...
4
votes
0answers
147 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 ...
15
votes
3answers
353 views

Local Inverse Galois Problem

It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedian field $K$ has solvable (in fact supersolvable) Galois group $G$. One sees this by using the ramification filtration ...
1
vote
1answer
225 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 $p$, $K^{sep}(not \ p)$ ...
2
votes
0answers
173 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 $\psi_p$, ...
1
vote
0answers
179 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 ...
1
vote
0answers
44 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 ...
9
votes
1answer
333 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 ...
4
votes
1answer
301 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 $U_a(K)$ ...
0
votes
0answers
72 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 ...
4
votes
1answer
188 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
vote
0answers
181 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 ...
7
votes
0answers
217 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 ...
1
vote
1answer
203 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 $F_{\infty}\subset \bar{F}$ ...
4
votes
0answers
353 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 ...
4
votes
1answer
153 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 ...
4
votes
1answer
195 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 ...