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Questions tagged [local-fields]

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Quadratic extension of local field

Let $F$ be a nonarchimedean local field of characteristic zero, and $E$ an extension of $F$ with $[E:F]=2^n$ for some $n$. Is it always possible to find a quadratic extension $M$ of $F$ such that $F\...
Windi's user avatar
  • 833
6 votes
1 answer
285 views

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 ...
Piotr Pstrągowski's user avatar
3 votes
1 answer
192 views

Frobenius-Schur indicator of a self-dual L-parameter

Let $F$ be a non-archimedean field and let $\pi$ be a self-dual supercuspidal representation of $\mathrm{GL}_n(F)$ (which, by a result of Adler exists only when $n=1$ or $n$ is even). Then, under LLC ...
Kenta Suzuki's user avatar
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1 vote
1 answer
264 views

How often does $-1$ have a square root in a local field?

Let $F$ be a nonarchimedean local field, say, charactersitic $0$. Is there any general theorem that tells when $\sqrt{-1}$ exists in $F$? How often does it happen?
Windi's user avatar
  • 833
58 votes
9 answers
16k views

Learning Class Field Theory: Local or Global First?

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...
David Corwin's user avatar
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5 votes
1 answer
493 views

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 $K/F$, ...
Sebastian Monnet's user avatar
0 votes
1 answer
146 views

Does an affine building associated to a group satisfy the axioms of building?

Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
M masa's user avatar
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1 vote
0 answers
180 views

Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$

$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $ Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...
user267839's user avatar
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3 votes
1 answer
296 views

$p$-power torsion of semiabelian variety

Let $K$ be a finite extension field of $\mathbb{Q}_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $...
Desunkid's user avatar
  • 247
2 votes
1 answer
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Vanishing of the degree 2 cohomology of a p-adic field with coefficients Q/Z and action of the Frobenius and the Pontryagin dual of the inertia

Let $K$ be a $p$-adic field with Galois group $G$ and inertia subgroup $I\subset G$. Denote $(-)^\ast=\mathrm{Hom}_{cont}(-,\mathbb{Q}/\mathbb{Z})$. Using Tate local duality, we can compute $$H^2(G,\...
Adrien MORIN's user avatar
28 votes
3 answers
2k views

What is a tamely-ramified Weil-Deligne representation?

Let $W_F$ denote the Weil group of a finite extension of $\mathbb{Q}_p$. Let $I$ denote the inertia subgroup and $I^{>0}$ the (pro-$p$) subgroup of wild inertia. (I hope I've got my notation right.....
Geordie Williamson's user avatar
9 votes
1 answer
331 views

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 ...
Julien's user avatar
  • 163
4 votes
1 answer
313 views

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. ...
Suzet's user avatar
  • 769
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 ...
user38495's user avatar
  • 1,062
3 votes
0 answers
128 views

Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
Piotr Pstrągowski's user avatar
42 votes
2 answers
5k 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 ...
Stiofán Fordham's user avatar
13 votes
1 answer
765 views

Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split?

Let $K$ be a henselian valuation field with residue field $k$, then the decomposition group surjects onto Galois group of the residue field, with kernel the inertia subgroup, namely we have short ...
user avatar
9 votes
1 answer
322 views

A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...
Mihir Sheth's user avatar
2 votes
0 answers
128 views

Lubin--Tate formal group construction in local class field theory using group cohomology

Let $K$ be a non-archimedean local field of characteristic 0. Fix a uniformiser $\pi$ and an algebraic closure $\bar{K}$. The theory of Lubin--Tate formal groups gives an explicit construction of the ...
User0829's user avatar
  • 1,428
0 votes
0 answers
78 views

Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$

Let $K$ be a local field of positive characteristic. I'm looking for a $K$ which satisfies the following condition. Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n)...
Duality's user avatar
  • 1,541
4 votes
3 answers
219 views

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 ...
Windi's user avatar
  • 833
3 votes
0 answers
86 views

Cohomology of compact open subgroups of semisimple groups over local fields

Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
naf's user avatar
  • 10.5k
5 votes
0 answers
181 views

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 \...
Mark OSS's user avatar
  • 159
1 vote
0 answers
256 views

Globalization of a local field

I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1. Here is the statement. ...
user avatar
6 votes
1 answer
424 views

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 \...
Lios's user avatar
  • 213
4 votes
1 answer
347 views

A Tate-Sen theorem mod $p$

Let $p$ be a prime number, $G=\textrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and $\chi:G\rightarrow\mathbb{Z}_p^\times$ the cyclotomic character. Let $\mathbb{C}_p$ denote the completion of the ...
jacob's user avatar
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2 votes
0 answers
338 views

Existence of "nth root function" which is analytic

Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...
user474's user avatar
  • 123
4 votes
1 answer
507 views

Galois cohomology of separable closure

Let $K$ be a local field, $K^{sep}$ its separable closure, $G = Gal(K^{sep}/ K)$ the Galois group and $C := \overline{K^{sep}}$ the completion with respect to the induced valuation. In his paper on $p$...
Piotr Pstrągowski's user avatar
1 vote
1 answer
284 views

$K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$

Let $K$ be a number field and $v$ be it's one of $K$'s non-archimedian valuation. Then, I would like to prove $K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$. This is from Silverman's ...
Duality's user avatar
  • 1,541
1 vote
0 answers
88 views

The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions

Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map \begin{equation} \mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
A. Maarefparvar's user avatar
3 votes
0 answers
117 views

Why inherit the Tits systems structure by a $B$-adapted homomorphism?

Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...
M masa's user avatar
  • 479
3 votes
0 answers
191 views

Decomposition of primes in cyclotomic extensions and their ramifications

Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$. So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...
user100603's user avatar
4 votes
3 answers
638 views

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?
Dipendra Prasad's user avatar
1 vote
0 answers
169 views

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$?

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ? I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...
Duality's user avatar
  • 1,541
2 votes
0 answers
99 views

Are tamely ramified representations $\widehat{\mathbb{Q}_p^\text{tr}}$-admissible?

Let $K$ be a finite field extension of $\mathbb{Q}_p$. Let $G_K$ denote the absolute Galois group of $K$, $I_K$ the inertia subgroup and $I_K^{(p)}$ the $p$-Sylow subgroup of $I_K$, i.e. the wild ...
Konstantin's user avatar
2 votes
1 answer
141 views

Reference to basic facts on non-Archimedean local fields

I need a reference to the following claims which, I believe, are correct and well known to experts (I am not one of them). Let $K$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of ...
asv's user avatar
  • 21.8k
1 vote
0 answers
165 views

When is the extension $L(S)/L$ Galois and totally ramified?

Let $L$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ with uniformizer $\pi$. Let us consider the polynomial ring $L[x_1,x_2,\dotsc,x_l]$ in $l$-...
MAS's user avatar
  • 930
3 votes
1 answer
188 views

Is the set of hyperelliptic curves with a K-point closed?

I am actually interested in the same question for more general kinds of curves, but I will be specific. Let $K$ be a field and $\overline{K}$ be an algebraic closure of $K$. Let's say that a "...
Erik Walsberg's user avatar
3 votes
2 answers
500 views

A question related to supercuspidal representations of $\operatorname{GL}_2$ over local fields

I was learning about the representation of $\operatorname{GL}_2$ over local fields and came to know something like: if the residual characteristic of the local field is an odd prime, then every ...
user15243's user avatar
  • 424
6 votes
1 answer
219 views

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 ...
frafour's user avatar
  • 435
1 vote
0 answers
132 views

A question about Theorem 2.3.1 in Tate's thesis [closed]

I don't understand how to prove a conclusion in the Theorem. When k is $p$-adic, the subgroups 1+$p^{v}$, $v>0$, of $u$ $(|u|=1)$ form a fundamental system of neighborhoods of $1$ in $u$, We must ...
Fuutorider's user avatar
22 votes
2 answers
1k views

Langlands correspondence for higher local fields?

Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible ...
user avatar
2 votes
0 answers
88 views

IS the composition of infinite APF extensions again APF?

Convention: By APF extension, I mean APF extension of $\mathbb{Q_p}$. For $\mathbb{Q_p} \subseteq L_1 \subseteq L_2$ where $L_2/L_1$ is finite, we know that $L_1/\mathbb{Q_p}$ is APF iff $L_2/\mathbb{...
Ehsan Shahoseini's user avatar
22 votes
3 answers
2k views

Totally ramified subextension in a finite extension of $\mathbf{Q}_p$

Let $K$ be a finite extension of $\mathbf{Q}_p$. Let $F_d$ be the unramified extension of $\mathbf{Q}_p$ of degree $d$. I would like to know whether there exists some $d \geq 1$ and some $L \subset K \...
Laurent Berger's user avatar
4 votes
2 answers
1k views

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 ...
Teppi's user avatar
  • 91
2 votes
0 answers
141 views

Refinement 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 $K/F$, ...
Sebastian Monnet's user avatar
2 votes
0 answers
731 views

On Serre's "Local fields"

While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to ...
rime's user avatar
  • 445
5 votes
1 answer
223 views

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 ...
Keivan Karai's user avatar
  • 6,224
5 votes
0 answers
194 views

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^{\...
M masa's user avatar
  • 479
2 votes
1 answer
127 views

The notion of smoothness in the local situation

I am reading Bump's book on Automorphic forms and Representations and I am able to draw a lot of parallels between the theory of $GL(2, \mathbb{R})$ which is the infinite place and the theory of $GL(2,...
Krishnarjun's user avatar