Questions tagged [local-fields]
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262 questions
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is there a unique measure on a local field?
Suppose we consider a local field and forget about the topology for a moment and consider the set of all measures on some non-trivial $\sigma$-algebra over the field that makes the field operations ...
- 2,125
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272
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sequences in non-abelian group cohomology
In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
- 273
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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
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253
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Under What assumptions on $p$, $\mathcal{O}_K^* \simeq \mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$
Let $p$ be a fixed prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers and $K$ be an extension of degree $2$ of $\mathbb{Q}_p$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\...
- 101
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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
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462
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Complete fields with algebraically closed residue field
I am looking for a reference where the following result is proven:
Let $k$ be an algebraically closed field. If $K$ is a complete and discretely valued field with residue field $k$. Then $K$ is one ...
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216
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Factorisation of polynomials over finite field
Is there a method to factorise a polynomial, for $k \leq m$ and $a_i \in \mathbb{F}_p$,
$$
1 + t^k(1 + a_1 t + a_2 t + \ldots + a_m t^m)^k
$$
as a product $$
(1 + t^k)^{x_1} \cdots (1 + t^l)^{x_l} \...
- 601
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654
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Dyadic Hilbert symbols and higher unit groups
Let $F$ be a local dyadic number field, $\mathfrak{p}$ its maximal ideal, $(*,*)_F$ its quadratic Hilbert symbol and $e$ its ramification index (i.e. $\mathfrak{p}^e$ is exact divisor of $2$). Fix an ...
- 101
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600
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A formal group scheme in explicit local class field theory
Let $K$ be a nonarchimedean local field with residue field $k$ of characteristic $q = p^N$, and pick a uniformizer $\pi\in \mathscr{O}_K$. Recall that explicit local class field theory, à la Lubin--...
- 5,770
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Decomposition of $\widehat{k^{\times}}$ occuring in local class field theory
Let $k$ be a finite extension of $\mathbb{Q}_p$ very often we use the isomorphism that $Gal(\overline{k}/k)^{ab} \simeq \hat{(k^{\times})}$ given by local class field theory.
My question would be do ...
- 385
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Subfields of higher local fields
Q1. Let $K$ be a local field with valuation $v$. Let us call $K'\subset K$ a nice local subfield if it is complete with respect to the induced from $K$ valuation. By local subfield I will mean a ...
- 2,305
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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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495
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Characters of the kernel of the norm map of an extension of local fields
Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm ...
- 41
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Is a reductive group scheme always parahoric?
Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
- 407
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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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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 ...
- 2,799
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478
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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 ...
- 980
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167
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$p$-primary torsion of an elliptic curve in the cyclotomic $\mathbb{Z}_p$-extension of a $p$-adic field
Let $K$ be a number field and $v$ be a fixed prime above $p$. Let $k=K_v$. We have the cyclotomic $\mathbb{Z}_p$ extension $K_\infty/K$ and if $w$ is a prime above $v$ in $K_\infty$ we write $k_\infty=...
- 1,283
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3
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523
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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.(...
- 61
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851
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Understanding the structure of unitary groups
I would like to understand precisely the structure of unitary groups.
Let $F$ be a global number field, $E$ a quadratic extension of $F$, and $U$ a unitary group on $E$ (i.e. the group of ...
- 5,424
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370
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The Breuil-Mezard Conjecture and Generalizations (Survey)
What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?
- 171
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Finding a cyclic cubic extension of a field
Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...
- 1,017
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414
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Why is $\mathbb{Q}_p(p^{1/p^\infty})$ a complete topological field?
In Matthias Wulkau's exposition of Scholze's thesis, the term perfectoid field is defined as follows:
Let $K$ be a field endowed with a non-archimedian absolute value $\lvert\cdot\rvert$, and let $\...
- 1,247
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470
views
Explicit $H^2(K, \mu) = Q/Z$?
In the development of local class field theory, a very fundamental theorem is that, for every local field $K$ of characteristic zero,
$H^2(K, \mu) \cong \mathbb{Q}/\mathbb{Z}$. $(*)$
Neukirch et al. ...
- 980
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347
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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
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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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370
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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 ...
- 3,625
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Is the special case of Abhyankar's lemma is also considered as such?
Consider the following statement:
Assume $E$ and $F$ are unramified (over some fixed prime) finite separable extensions of a field $K$. Then $EF$ is also unramified.
I always thought that it is ...
- 3,362
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375
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compact Zariski-dense subgroups of p-adic groups
Given an algebraic group $G$ defined over a $\mathbb Q_p$. It has two topologies: one is induced by the $p$-adic metric, the other is the Zariski topology. Let $C$ be a compact (w.r.t. the $p$-adic ...
- 453
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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 $\...
- 121
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Inverse image of norm map on principal units for an unramified extension
For a local field $E$, denote by $U(E)$ the units of the corresponding valuation ring $\mathcal{O}_E$, and denote by $U_n(E)$ the prinicipal $n$-units, i.e. $U_n(E)=1+M_E^n$ where $M_E$ is the maximal ...
- 225
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Reference for: Every local field can be realized as the completion of a global field
It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...
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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
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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, ...
- 91
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99
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Monoid cohomology of $\mathbb{N}$ for a linear algebraic group
Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
- 273
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526
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Integral representation of adjoint L-factor for GL(2)
My question is about a local computation in the paper of Gelbart and Jacquet, "A relation between automorphic representations of GL2 and GL3", from 1978.
Let $\sigma$ be an irreducible smooth complex ...
- 37k
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516
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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 ...
- 71
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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
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2
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Image of norm map for local field
Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$.
What is the image of the norm map $N_{E/F}$?
In particular - ...
- 899
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409
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Higher Adeles of a scheme and related topics
Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar).
Consider the augmented ...
- 7,799
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1
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190
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Hilbert symbols vanishing
Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitve $p^2$-th root of unity $\zeta_{p^2}$. Let $a,b \in E^...
- 11.3k
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Local triviality of Galois cohomology classes over $\mathbb{Q}$
Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in
$$\mathrm{H}^1(\mathbb{...
- 21.4k
3
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2
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340
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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 ...
- 31
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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
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415
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Locally compact vector space over a finite field
In the wikipedia article titled "topological vector space", there is a line saying the following.
"Let $K$ be a locally compact topological field, for example to real or complex numbers. A ...
- 540
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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
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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 $U_a(K)$ (...
- 899
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3
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1k
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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 \...
- 2,824
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Fields of definition of parabolically induced representations of $\mathrm{SL}(2,q)$
Let $\alpha_0$ be the unique non-trivial character satisfying $\alpha_0^2=1$ of the split torus $\mathrm{T} \subset \mathrm{SL}(2,q)$ and denote by $\mathrm{R}(\alpha_0)$ the character of $\mathrm{SL}(...
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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?
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