Questions tagged [local-fields]
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262 questions
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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 ...
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Are the abelian absolute Galois groups of these local fields isomorphic?
For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$.
Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) \...
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Finite extension of local fields
Can a (higher) local field have uncountably many finite (seperable) extensions?
- 11.3k
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Is there any explicit description of the maximal totally ramified extension of $\mathbb{Q}_p$?
It is well known that the maximal unramified extension of $\mathbb{Q}_p$ can be extended by adding the roots of unity of order prime to $p$. Is there any explicit description of the maximal totally ...
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Is every compact simply-connected reductive p-adic group perfect?
Let $k$ be a nonarchimedean local field and $G$ a reductive $k$-group,
which we assume to be semisimple and simply-connected. Recall that an abstract group $H$ is perfect if it is generated by ...
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Why is $K_{\upsilon}|K$ separable for a global field $K$?
I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...
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3
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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 $k_v$.
Proposition.(...
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Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?
Let $K$ be a $p$-adic local field, for example $\mathbb{Q}_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$($v\ge -1$) be the ramification groups in upper numbering, then is it true ...
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Can every finitely generated field extension of $\mathbb{Q}$ be embedded into a local field?
Let $K$ be a finitely generated field extension of $\mathbb{Q}$, and let $p$ be a prime number. Can $K$ must be embedded into a p-adic local field (i.e. a finite field extension of $\mathbb{Q}_p$) ?
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Integral Tate-Sen theory
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...
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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 $I_p$, ...
- 11.3k
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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 $U_a(K)$ (...
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Galois module theory: from global to local
Let $L/\mathbb{Q}$ be a finite Galois extension with Galois group $G$. It is well known that the ring of integers $\mathcal{O}_L$ is free over its associated order $\mathfrak{A}_{L/\mathbb{Q}}=\{x\in \...
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Generating the coordinate ring of the Lubin-Tate formal group
Let $K$ be a $p$-adic local field with uniformizer $\pi \in \mathcal{O}_{K}$ and residue field $k = \mathcal{O}_{K}/\pi$. Let $G$ be a Lubin-Tate formal $\mathcal{O}_{K}$-module and $G_{0}$ its ...
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Cohomology of finite $p$-groups over integers in local fields
Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2).
In ...
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Extensions of p-adic number fields
Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...
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Abelianization of the inertia group
Let $F/\mathbb Q_p$ be a finite extension, and let $I_F=\operatorname{Gal}(\overline F/F^{\mathrm{unr}})\subset\operatorname{Gal}(\overline F/F)$ be the inertia subgroup.
Is there a description of ...
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$SL_2(\mathbb{Z}_p)$ extension of a local field
Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...
user141691
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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 ...
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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-...
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What are the jumps in the ramification filtration of the absolute Galois group of a local field?
Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...
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Is there any approximated version of Hilbert 90?
Suppose $K$ is a local field and $L$ a finite cyclic extension of $K$. By Hilbert 90, we know that if an element $a$ in $ L$ such that $N_{L / K}(a) =1 $ then $ a = b / \sigma(b) $ for some $b$ in $L$ ...
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Looking for proof of Serre's mass formula
Let $F$ be a finite field extension of the $p$-adic numbers $\mathbb{Q}_p$, whose residue field has $q$ elements. Let $\mathfrak{p}$ be the prime ideal of $F$. Given a finite field extension $K/F$, ...
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A question on linear algebra over non-Archimedean local field
Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it ...
- 21.8k
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Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
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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 ...
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When is $G(R)\rightarrow G^{\textrm{ad}}(R)$ surjective?
Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The ...
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Torsion subgroup of the group of points of an elliptic curve over local field
Let $K$ be a local field with residue field $k$ and $E/K$ an elliptic curve. I'm interested for which $K$ and $E$ the group of torsion points on the curve is finite. I can prove that this group is ...
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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 ...
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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 ...
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Regular elliptic elements are dense in p-adic division algebra
I'm trying to better understand the set $E$ of regular elliptic elements of $D^\times$, where $D$ is a finite dimensional central division algebra over a non-archimedean local field $F$.
For example, ...
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Artin map restricted to base field
Let $M/L/K$ be a tower of local fields such that $M/L$ is abelian with Galois group $G$. The Artin map $\psi_{M/L}$ restricted to $K^\times$ is a continuous map to $G$ and thus corresponds to some ...
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Relation in Brauer group coming from trace form
Let $L/K$ be a cubic (or, more generally, odd-order) extension of fields of characteristic $0$. To every element $a \in L^\times$ we can associate the quadratic form
\begin{align*}
q_a : L &\to K \...
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Question on the unramified local Langlands conjecture
I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or ...
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defining the upper ramification numbering
Given a local field $K$ with absolute Galois group $\Gamma$. Is it "possible" to define the upper numbering on $\Gamma$ without using the lower numbering?
In other words, given $\gamma \in \...
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Haar mesure on $\mathrm{GL}_{d}(F)$
$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...
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What is $H^1(\mathbb Z, GL_k(R_n))$ for a ring closely related to the cyclotomic rings of integers?
Let us consider $R_n = \mathbb Z_\ell[\theta_n]/(\theta^{\ell^n}-1)$, an auxiliary prime power $q\equiv 1 \pmod \ell$ with an action of $\mathbb Z = \langle \sigma\rangle$ by $\sigma(\theta_n) = \...
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Explicit construction of abelian wild inertial extensions of maximal tamely ramified extension of $\mathbb{Q}_p$?
In Iwasawa's paper On Galois groups of local fields, he proves that if $V$ is the maximal tamely ramified extension of $\mathbb{Q}_p$, with Galois group $\Gamma$ over the base, then its abelianized ...
- 2,054
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Arithmetic of Cuspidal Reps. Fundamental non split stratum and simple stratum
I started to read Colin Bushnell's notes on this title. The last theorem in the 3rd section claims that if $\pi$ is an irr. smooth rep. of $GL_N(F)$ containing a non-split fundamental stratum, then $\...
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Topology on two dimensional local fields
I posted my question here, but there is no reply yet. So, I guess I should post it on mathoverflow.
I am reading the book of Schneider about Galois representation and $(\varphi,
\Gamma)$-module, ...
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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
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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,...
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Why isn't there a structure with two primes?
I don't know whether this question is a bit too vague for MO or not, so feel free to delete it if you see fit.
The p-adic integer is defined by taking the inverse limit $\ldots \mathbb{Z} / p^2 \...
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Reference request for Kato's paper: A generalization of local class field theory by using K -groups
I would like to ask for the paper of Kato: A generalization of local class field theory by using K -groups I, J. Fac. Sci. Univ. Tokyo Sec. IA 26 No.2, 1979, 303–376. I could not find it. Could anyone ...
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Possible $p$-torsion subgroup of $E(\mathbb{Q}_p)$, and if there is a theorem to say which case happens when?
What is the possible $p$-torsion subgroup of $E(\mathbb{Q}_p)$ for an elliptic curve $E$ over $\mathbb{Q}_p$, and if there is a theorem to say which case happens when?
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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?
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Is $K^\times/ F^\times$ compact for local fields?
Let $K/F$ be a finite extension of local fields (of characteristic 0). Is it true that the quotient group $K^\times/ F^\times$ is always compact?
I understand that if the extension is cyclic, it is ...
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593
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Hilbert Symbols, Norms, and p-adic roots of unity
Let $p$ be an odd prime number,
let $\mathbb{Q}_p$ be the field of $p$-adic numbers,
and let $\overline{\mathbb{Q}_p}$ be an algebraic closure of it.
For a primitive $p$-th root of unity $\zeta_p \in ...
- 11.3k
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1
answer
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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
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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 ...
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