Questions tagged [local-fields]
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262 questions
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Unramified Galois cohomology
Let $k$ be a local field with absolute Galois group $\Gamma_k$, inertia subgroup $I_k \subset \Gamma_k$, and residue field $\mathbb{F}$. Let $M$ be a finite Galois module over $k$.
The unramified ...
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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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Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that
The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
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Zero of power series and Newton polygon in non-archimedean complete algebraically closed fields
In Gouvea book $p$-adic numbers, on can find this corollary (7.4.11)
Let $f(X) = 1+a_1X+a_2X^2+a_3X^3+\cdots$ be a power series which converges on the closed ball of radius $c = p^m$. Let $m_1, m_2, \...
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Can two elements always belong to the same Laurent series field?
Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$ on $\mathbb F_q(T)$.
Let $x,y\in\overline{\...
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Zero of a power series in a local field
Let $f(z)=\sum_{n\ge1}a_nz^n$ be a power series of $\mathbb C_p[[z]]$ where the $a_n$ are such that $|a_n|=1$ for every positive integer $n$. Consider $z_0\in\mathbb C_p$ such that $|z_0|<1$. Can ...
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Ring structure on Brauer group
Class field theory defines an isomorphism between the Brauer group of a finite extension of p-adic fields and a cyclic group with a canonical generator. This in turn defines an isomorphism of the ...
CommunityBot
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Time complexity of Magma's `NormEquation` for quadratic extensions of $2$-adic fields
Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...
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Is the exponent of $2$ in the Pythagorean theorem the "same $2$" as $[\mathbb{C} : \mathbb{R}]$?
I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...
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Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
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is a linear algebraic group over an extension of $\mathbb{Q}_p$ a locally pro finite group?
Let $F$ be a non archimedean local field and let $G$ be linear algebraic group over $F$. I do not have a lot of experience with linear algebraic group, but it seems very obvious that $G$ inherits the ...
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References for Bernstein-Zelevisnky classification
I am looking for references for the Bernstein-Zelevisnky classification of irreducible representations of GL$(n,F)$ in terms of cuspidal representations. In particular I would like to find something ...
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Galois Cohomology mod 2 of iterated Laurent series
Let $k$ be an algebraically closed field of characteristic different from two.
For $n\geq 1$, set $F_n=k((X_1))\cdots ((X_n))$, and let $F=\displaystyle\bigcup_{n\geq 1}F_n$.
If $I=\{i_1,\ldots,i_m\}$...
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Ramification at particular level of a tower of extensions of local field
Let $K$ be an unramified extension of the $p$-adic number field $\mathbb{Q}_p$.
Suppose we have a tower of extensions:
$$K=:K(u_0) \subset K(u_1) \subset K(u_2) \subset K(u_3) \subset \cdots \subset ...
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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 ...
CommunityBot
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Local Class field theory and Artin map for the Weil group
I am searching a reference for local class field theory that use the Weil group instead of the absolute Galois group. In particular that the Artin map is an isomorphism between the multiplicative ...
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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$?
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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 ...
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Regarding upper numbering of ramification groups
In Serre's book "Local fields" he defines the function $\phi(u)=\int_{0}^{u}\frac{dt}{( G_0:G_t)}$ and defines the upper number of ramification groups as $G^v=G_{\phi^{-1}(v)}$ and somehow ...
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Is there any relationship between the study of class number of a number field with the study of class field theory through Lubin-Tate formal group?
I am curious to know if we can somehow relate to the study of local class field theory through Lubin-Tate formal group with the study of class number of a field (global field in general) in class ...
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$p$-adic counterpart of W-algebra
Representation theory and geometry over $k((t))$ and $\mathbb{Q}_p$ have many similarities, and there are many similar constructions, usually motivated from the other side (say the study of affine ...
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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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Tensor product of finite extensions of $\mathbb{Q}_p$
Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.)
$(1)$ If $M$ is a finite Galois extension of $F$ with Galois ...
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Topology of multiplication groups of local fields
In Neukirch’s book “Algebraic Number Theory”, Proposition II.5.7, the following is insisted: for a mixed characteristic local field $K$ with a residue field $\mathbb{F}_q$, $q = p^f$, then one has an ...
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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 ...
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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 ...
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields
Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
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Conductor and local Kronecker–Weber theorem
Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...
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Compact subgroups of a linear group over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
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Some questions about splitting of sequence $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ for Henselian val field $K$
I have a couple of questions about following proof by Peter Scholze on splitting of the ses (...does it have a name?...)
$$0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$$
for $K$ henselian valuation ...
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Question on the unramified local Langlands conjecture
I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or ...
- 12.9k
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$n$-th root of character on local field
Let $F$ be a non-Archidean local field of characteristic 0, and $\zeta_n$ the set of $n$-th roots of unity in the algebraic closure of $F$. Assume $\zeta_n\subseteq F$. Let $\chi:F^\times\to\mathbb{C}^...
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Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields
Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...
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Existence of tamely ramified tower of extension over $\mathbb{Q}_p$
Let $p$ be a prime. There exist following containment : $$\mathbb{Q}_p \subset \mathbb{Q}_p^{\rm nr} \subset \mathbb{Q}_p^{\rm tr} \subset \overline{\mathbb{Q}}_p$$
Here $\mathbb{Q}_p^{\rm nr}$ and $\...
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Unramified composition for every extension
Let $K$ be a number field and $S$ be a finite set of primes. Is it possible to construct a finite extension $M$ of $K$ such that $LM/M$ is unramified at (the primes above) $S$ for all degree $n$ ...
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Local systems on $\mathbb P^1$ and on the formal punctured disc
Consider the projective curve $\mathbb P^1$ over a finite field $k$.
Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that
a) $E$ is tame at $\infty$
b) The ...
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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 $R$ ...
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For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
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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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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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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 ...
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Completion of $\mathbb F_q(T)$
It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...
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Tempered Iwahori-spherical representations
Consider a local field $F$ of characteristic 0 and $G=GL_n(F)$.
It is well known (for example Cartiers article in Corvallis) that an admissible, irreducible representation $\pi$ of $G$ has a ...
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Why does the field norm on the field extension $ \mathbb C/\mathbb R $ induce a vector space norm?
There is a general result which holds for the rational numbers $ \mathbb Q $ (as well as number fields in general):
For any completion $ K $ of $ \mathbb Q $ and any finite extension $ L/K $ of ...
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Confusion with self-dual representations of $\mathrm{GL}_n$ over a $p$-adic field
The following surely is kind of a trivial question, but it keeps me confused. It concerns a detail in Lust and Stevens' paper "On depth zero L-packets for classical groups" London Math. Soc. ...
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Cohomology of local fields in positive characteristic
It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
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Given $v,w$ primes of $k$, is there $K/k$ so $K_v\cap\Bbb Q^\text{cycl}=K_w\cap\Bbb Q^\text{cycl}=K\cap\Bbb Q^\text{cycl}$?
For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there ...
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Can global fields be defined as certain topological fields like local fields?
It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...
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What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2 \mathbb{Z}_p$?
Let $p \geq 2$ be prime and $K=\mathbb{Q} (\zeta_p),~ \zeta^{p-1}=1$ with ring of integers $\mathcal{O}_K$. We denote $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ ...
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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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