Questions tagged [class-field-theory]

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185 views

What is the conductor of $K(\sqrt{2})$ over $K$?

Let $ K=\Bbb Q\left(\sqrt{(-1)^\frac{N+1}{2}N}\right)$. I want to find the ray class field of $K$ containing $\sqrt{2}$. I considered $L=\Bbb Q(\sqrt{2})$. By Artin reciprocity, there exist modulus $\...
17 votes
4 answers
1k views

What's the use of group cohomology for class field theory?

I'm a graduate student studying now for the first time class field theory. It seems that how to teach class field theory is a problem over which many have already written on MathOverflow. For example ...
4 votes
1 answer
139 views

Class numbers of cyclotomic fields and their maximal totally real subfields

Let $\zeta_p$ be a $p$-th root of unity for a prime $p$, let $L:=\mathbb{Q}(\zeta_p)$ and $K$ the maximal totally real subfield of $L$, i.e. $K:=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$. I am trying to prove ...
  • 409
2 votes
0 answers
109 views

relating class number and narrow class number of a real field

I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...
  • 409
2 votes
1 answer
59 views

Ramification of primes and order of $\smash{\hat{H}}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of ...
8 votes
2 answers
307 views

Image of the norm map for Artin-Schreier extensions

Let $k$ be a local field of characteristic $p$ and $\omega \in k$ a uniformiser. Consider the Artin-Schreier extension $L_n = k[x]/(x^p - x - \omega^n)$ for each $n \in \mathbb{Z}$. Is there an ...
2 votes
0 answers
98 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 ...
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1 vote
0 answers
129 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 ...
1 vote
0 answers
75 views

CM-fields and ideal theoretic ray class group

Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $S$ a finite subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}_K$ an ideal. The ideal theoretic ray class group of $\...
1 vote
2 answers
124 views

The $ 1 \operatorname{ mod } \mathfrak{m}$ congruence relation in ray $ P^{\mathfrak{m}}$ of the ideal theoretic ray class group

Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $S$ a subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}_K$ an ideal. The ideal theoretic ray class group of $\...
2 votes
1 answer
166 views

A question about unramified quadratic extension of number field

Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?
3 votes
0 answers
57 views

Finiteness for Galois cohomology for $\mathbb{Z}_p$-module coefficients

I am looking for a general survey on the finite generation properties of $$H^i(F,\mathbb{Z}_p(j))$$ for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is ...
3 votes
1 answer
118 views

Knot group of a field extension

Notation: $L/K$, finite extension of global fields $K^\times$, unit group of $K$ $L^\times$, units group of $L$ $\mathbb{A}_L^\times$, ideles of $L$ $N_{L/K}$, the norm map The knot group of an ...
4 votes
1 answer
224 views

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 ...
0 votes
1 answer
128 views

Class number of imaginary quadratic fields

Let $n$ be a positive squarefree integer, and let $h_n$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$. Then, is it true that $h_n$ is odd if and only if $n$ is a ...
2 votes
0 answers
83 views

Artin reciprocity for function fields vs number fields

I have a naive question about the difference of Artin reciprocity in the number field versus function field case: In the number field case, the double quotient we look at is $$K^\times \backslash \...
2 votes
0 answers
83 views

Rayclass group and Hilbert class group, $\mathrm{Gal}(K(\mathfrak{a})/K)\cong(\mathcal{O}_K/\mathfrak{a})^\times$

Let $K$ be an imaginary quadratic field with class number $1$. Let $\mathfrak{a}$ be an ideal of $ \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$. Why ...
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-1 votes
1 answer
137 views

Set of all primes $p$ that split in $\mathbb{Q}\left(\sqrt{-k}\right)$

Let $k$ be a squarefree positive integer. We know that a prime $p$ splits in $K=\mathbb{Q}(\sqrt{-k})$ if and only if $-k$ is a quadratic residue mod $p$. My question is: can we explicitly determine ...
4 votes
1 answer
127 views

Existence of lift of (local) Artin map

In a comment to this question, David Loeffler asked if one can show that the (local) Artin map $$K^\times \to G_K^{ab}$$ does not have a lift to $G_K$. Probably this wouldn't be canonical, but I can't ...
2 votes
0 answers
56 views

Narrow class number of a the maximal totally real number field inside a cyclotomic field

I am wondering how much it is known about the narrow class number of the number field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ ($p$ an odd prime). More precisely, I am interested to know when it is odd. By ...
  • 409
5 votes
0 answers
130 views

Essence of relations between central simple algebras and Galois cohomology in canonical morphism of class field theory

I was somewhat puzzled after I finished learning class field theory for several times. My question is about the relations between "classical simple algebras, Brauer groups" and "modern ...
4 votes
0 answers
135 views

What are the applications of $\lambda$-rings to class field theory?

In the book Lambda Rings by Yau, he mentions several areas where $\lambda$-rings can be applied, but he doesn't go into much details. He even includes class field theory in the list, mentioning "...
1 vote
0 answers
113 views

Units in residue classes modulo prime ideal

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}...
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2 votes
0 answers
126 views

About Lubin Tate extensions

https://math.stackexchange.com/questions/4473761/about-lubin-tate-extensions I know this is a very low-level question because it is about a probable typo and a definition. But I asked this question on ...
2 votes
1 answer
125 views

Constructing Ray class fields of $\mathbb{Q}(i)$ and division points of lemniscate

I've heard that it was Abel who first constructed some of (or all of?) Abelian extensions of $\mathbb{Q}(i)$ using division points of lemniscates. Can we construct the Ray class fields of $Q(i)$, or ...
5 votes
1 answer
230 views

Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?

For $K$ a number field, denote by $\mathcal{O}_K$ its ring of integers and by $H_K$ its Hilbert class field. For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined ...
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3 votes
1 answer
257 views

Lang's proof concerning ray class fields of imaginary quadratic number fields

Crosspost from Math.SE as I did not receive an answer there: In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $...
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6 votes
0 answers
188 views

Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$

$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
6 votes
1 answer
403 views

Chebotarev density theorem and pure weight local systems

How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper. Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...
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2 votes
0 answers
121 views

Artin map and profinite completion of the idèles

One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...
0 votes
0 answers
112 views

Tate cohomology and cup product: functoriality in $G$

Let $G$ be a finite group, and let $A, B$ be $G$-modules. See Atiyah and Wall [AW] for the definition of the Tate cohomology groups $H^q(G,A)$ for all $q\in\mathbb Z$ and of the cup product pairings \...
2 votes
0 answers
92 views

Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)

Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$ Question: Given a ...
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0 votes
1 answer
254 views

Coboundary operators, 1-cocycles and computing cohomology

My question is about the compatibility and consistency between two definitions of cohomology in two books. I asked this question about 10 days ago on MathSE and I set a bounty on it, but I didn't ...
3 votes
3 answers
426 views

Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected?

I was reading this question The connected component of the idele class group but I am very confused about the structure of the solenoids $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\...
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5 votes
0 answers
300 views

The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field, and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$. By local class field theory, there is a canonical isomorphism $$...
7 votes
1 answer
235 views

Explicit cocycles for the first Galois cohomology of a $p$-adic torus

Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$). Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...
2 votes
0 answers
119 views

Conditions for being an entry in a trace compatible sequence

$\DeclareMathOperator\Tr{Tr}$Let $K$ be a local field and let $q$ be the size of the residue field of $K$. $\pi$ will be a uniformizer of $K$. Let $f(X) = \pi X + X^q$. Then there is a unique formal ...
8 votes
1 answer
232 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}(...
12 votes
1 answer
463 views

Finite Galois module whose Ш¹ is nonzero?

In algebraic number theory, we constantly make use of the nine-term Poitou-Tate sequence: Let $K$ be a number field and $M$ a finite $K$-Galois module. Then we have the nine-term exact sequence $$ H^0(...
4 votes
1 answer
224 views

How to calculate genus number of number field using sage?

I am looking to find real quadratic fields whose Hilbert class field is abelian over $\Bbb Q$. Then I learned about genus numbers and genus field of the number field. It is enough to find a number ...
2 votes
0 answers
89 views

Is there data base of quadratic fields which have abelian Hilbert class field?

I did not find it in LMFDB. However, I am looking for a Database; please feel free to navigate me to any paper, which might be helpful. I really appreciate any help you can provide.
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4 votes
0 answers
225 views

Explicit invariant map in local class field theory

Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map: $$ H^2(\...
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4 votes
1 answer
208 views

A Kummer exact sequence involving $\mu_\infty$

Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...
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2 votes
1 answer
131 views

Field extension corresponding to a quotient of units of local fields

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $\mathcal{O}$ be its ring of integers and $\mathfrak{m}$ the maximal ideal. Pick a uniformiser $\pi$. The construction using theory of Lubin--Tate ...
  • 1,222
4 votes
0 answers
208 views

Reference request for Kummer-Artin-Schreier-Witt theory

I cannot find the following 4 papers by Sekiguchi–Suwa in their works on Kummer–Artin–Schreier–Witt theory: On the unified Kummer–Artin–Schreier–Witt theory, Prépublications du laboratoire de ...
3 votes
0 answers
174 views

Simplification of links between idele class group and étale cohomology

I posted this question over on stack exchange and was told it would work better here. For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
3 votes
0 answers
170 views

Central division algebras over $ \mathbb{Q} $

Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...
  • 693
2 votes
0 answers
141 views

Construction of genus class fields

Given a finite extension $K/\mathbb{Q}$, the genus class field $L$ is defined to be the maximal abelian extension of $\mathbb{Q}$ that is a subfield of the Hilbert class field $H$ of $K$. I am trying ...
  • 549
4 votes
1 answer
175 views

Norm groups of number fields

I came across this proposition in an article about genus class fields. I have a few questions about the parts that I have underlined in red. I don't understand why the norm map $N_{H/K}: I_H \to P_K$ ...
  • 549
3 votes
1 answer
156 views

Centralizer of the absolute Galois group of a number field

By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...

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