# Questions tagged [class-field-theory]

The class-field-theory tag has no usage guidance.

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### Reference request: ramified and local geometric class field theory

There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...

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161 views

### 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 ...

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86 views

### What is the image of $-1$ by the local reciprocity map?

Consider the Weil group $W$ of $\mathbb{Q}_p$, that is, the subgroup of those elements of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ mapping to an integer power of Frobenius. Class field ...

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211 views

### Easy cases of Herbrand's theorem

$\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$ I recall Herbrand's theorem about class groups of cyclotomic fields: Let $p$ be an odd prime, let $\zeta$ be a primitive $p$-th root of $1$ and let $K = \QQ(\...

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157 views

### Class fields without class field theory

Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...

**7**

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**1**answer

206 views

### Do we need the Weber function to generate ray class fields of imaginary quadratic fields of class number one?

I'm a bit confused by the role of the Weber function in generating ray class fields of imaginary quadratic fields of class number one. More specifically, let $K$ be such a field and $E$ an elliptic ...

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42 views

### Finite generation for a restricted ramification idele module

Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...

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64 views

### non $p$ part of the class group and analogous results

Washington had proven in 1978 that for $q$ a prime ($q \neq p$), if $q^{e_n}$ exactly divides the class number of $\mathbb{Q}_n$, ie the $n$-th layer in the cyclotomic $\mathbb{Z}_p$ extension, then $...

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123 views

### $p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?

Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...

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98 views

### When does a number field have $p$-rank greater than $n$?

Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\...

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**1**answer

157 views

### 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 ...

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**1**answer

218 views

### Are all totally ramified $\mathbb{Z}_p$-extensions of local fields come from (relative) Lubin-Tate formal groups?

The setup is as follows:
$k/\mathbb{Q}_p$ is a finite extension, $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_k$, $q=\#(\mathcal{O}_k/\mathfrak{p})$
$k'/k$ is a finite unramified extension of ...

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176 views

### What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...

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66 views

### Definition of the local Artin conductor for a representation of the Weil group

Let $\rho$ be a continuous, finite dimensional complex representation of the Galois group $\operatorname{Gal}(\overline{F}/F)$, for $F$ a $p$-adic field. Is there a general notion of an Artin ...

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196 views

### The closed subgroup of the idele corresponding to the maximal elementary $p$-extension of a global field

I want to know whether for any global number field $k$, the closed subgroup of the idele corresponding to the maximal elementary $p$-extension ($p$ is a prime number) is $k\ J^p$.
The critical point ...

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**1**answer

134 views

### Kummer congruences for totally real number fields

There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition here, more precisely see Theorem 2.1.
What is ...

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**1**answer

119 views

### $[J_F : P_FN_{E/F}J_E] = 2$ for quadratic extensions of global fields of characteristic 2?

Let $E/F$ be a quadratic extension of global fields. Denote by $J_F$ the idèle group of $F$, by $P_F$ the subgroup of principal idèles and by $N_{E/F} : J_E \to J_F$ the norm map between idèles. In ...

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85 views

### Reference for: power residue symbols are Hecke characters

Notation.
Let $n$ be a positive integer, let $\mu_n\subseteq \mathbb C$ be the set of $N$-th roots of unity, let $K$ be a number field containing $\mu_n$, let $R$ be the ring of integers of $K$, let $...

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155 views

### Restricted Iwasawa theory

Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...

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190 views

### What are the fastest ways to calculate class number of number fields?

Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$?
I am aware that the question is broad but any argument would be helpful.
Some basic approaches I know:...

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**1**answer

142 views

### Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?

If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute ...

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163 views

### Computing the relative class group (with Galois action) of relatively large cyclotomic groups

For a cyclotomic field $K = \mathbb Q(\zeta_n)$, let $K^+$ be its maximal totally real subfield. We know that $H^+ = Cl(K^+)$ injects into $H = Cl(K)$. I am interested in computing the group $H/H^+$ ...

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149 views

### Tables of class numbers of cyclotomic fields

Does anyone have a table of the class numbers ($h_n$) of cyclotomic fields (upto say, n = 250-300 for $\mathbb Q(\mu_n)$)?
I can find tables for the relative class number ($h_n^-$) in various places ...

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**1**answer

255 views

### Unramified non-abelian extension and Galois cohomology

Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class ...

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442 views

### 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--...

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**1**answer

216 views

### Brauer group of global fields

Is the Brauer group $\text{Br}(K)$ of a global field $K$
an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$?
Is $\text{Br}(K)[n]$ finite, for $n$ integer?
I know from class field ...

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76 views

### Abelian group extensions

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{...

**10**

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592 views

### What are the primes that are ramified?

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we ...

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**1**answer

266 views

### The Genus field and Hilbert class field

Is there an example of a number field $K$ for which the genus field of $K$ is contained strictly in the Hilbert class field of $K$?

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143 views

### Global sections of higher direct images

If $f : X\to V$ is a smooth proper map of smooth schemes, what are the global sections of
$R^if_{fppf, *}\mu_p$
$R^if_{fppf, *}\mathbb{G}_{\rm m}$
I was reading Milne's book "Arithmetic duality", ...

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94 views

### Fibers of reciprocity maps and higher dimensional analogs

Part I.
Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$.
We have the local Artin map for every finite $v$:
$$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...

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184 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. ...

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96 views

### Uniqueness of class field theory map

Let $F$ be a local field of characteristic 0. The main theorems in local class field theory can be summarized by the existence of a group $W_F$ and a map
$$
\phi_F:W_F\to W_F^\mathrm{ab}\simeq F^\...

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153 views

### On sum of squares?

If $(a,b)=1$ and $2|b$ then $p$ is prime and $p|a^2+b^2\implies p\not\equiv3\bmod4$.
For any other $k\in\Bbb N_{>2}$, is there a polynomial that represents an odd prime $p$ if and only if $p\not\...

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177 views

### On sum of squares representations of primes

We know $p=1\bmod 4$ has an unique representation via $x^2+y^2=p$ where $x,y\in\Bbb N$ holds.
There are other unique form representations as well.
Suppose $p$ is a prime and we have coprime $\alpha,\...

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**1**answer

185 views

### Examples of norm forms where the numbers represented can be readily described

In case of impatience: the question here is a request for examples, especially degree six or seven where the norm form might represent some prime$p,$ then some $q^2$ but not $q,$ then some $r^3$ but ...

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**1**answer

173 views

### Does Ribet's construction of class fields give us eigenspaces of rank 1?

Ribet's paper on the Herbrand-Ribet theorem constructs a representation $\rho: Gal(\overline{\Bbb Q}/\Bbb Q) \to GL_2(\mathbb F_q)$ where $q = p^r$ of the specific form:
$
\begin{bmatrix}
1 & *\\
...

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112 views

### 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 ...

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193 views

### correspondence between finite abelian extensions and congruence subgroups

I make self-study in class field theory and I want to prove the following popular fact:
Given a modulus $\mathfrak{m}$ of a number field $K$, the map $L\mapsto ker (\phi_{L/K,\mathfrak{m}}$) is an ...

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**2**answers

228 views

### The $\ell$- part of the class groups of the $p$-cyclotomic fields

Let $K_n = \Bbb Q(\mu_{p^{n+1}})$ and let $A_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A_n$. For instance, we know quite a lot about how it varies with $n$.
I am ...

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216 views

### Analogue of j-invariant for CM fields

For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic ...

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**1**answer

151 views

### 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^...

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640 views

### How to compute with the Stark conjectures?

I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to this MO question suggests that the Stark ...

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**1**answer

199 views

### 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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**1**answer

164 views

### CM Elliptic Curves and a result concerning ray class fields

Let $K$ be an imaginary quadratic field and suppose that $E/K$ has complex multiplication by $\mathcal{O}_K$. Let $\psi$ be the Hecke character associated with $E$ and $\mathfrak{f}$ its conductor (i....

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241 views

### 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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189 views

### Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result:
Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...

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82 views

### Quasi-algebraically closed fields reference request

I am looking for a road to understanding what quasi-algebraically closed fields are with the ultimate goal of understanding the paper by Lang 1952.
My current background is the first 6 chapters from ...

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**1**answer

319 views

### 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 ...

**2**

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**1**answer

263 views

### Determining the image of the norm map of a cyclic extension

Suppose $L / K$ is a cyclic extension of number fields. Is there a straightforward way to determine if a given $\alpha\in K^{*}$ is in the image of the norm map $N_{L/K}:L^{*}\rightarrow K^{*}$? Or to ...