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Questions tagged [class-field-theory]

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4
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150 views

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 ...
11
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0answers
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 ...
2
votes
0answers
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 ...
11
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0answers
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(\...
3
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0answers
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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1answer
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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0answers
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 $...
5
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0answers
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 $...
6
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0answers
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 ...
3
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0answers
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 $\...
2
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1answer
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 ...
6
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1answer
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 ...
10
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0answers
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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0answers
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 ...
3
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0answers
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 ...
4
votes
1answer
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 ...
2
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1answer
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 ...
3
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0answers
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 $...
4
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0answers
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 $...
6
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0answers
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:...
5
votes
1answer
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 ...
6
votes
2answers
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 $H/H^+$ ...
8
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0answers
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 ...
6
votes
1answer
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 ...
10
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0answers
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--...
1
vote
1answer
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 ...
1
vote
0answers
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
votes
1answer
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 ...
2
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1answer
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 $K$?
2
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0answers
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", ...
2
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0answers
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^{\...
7
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0answers
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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0answers
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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0answers
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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0answers
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,\...
4
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1answer
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 ...
8
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1answer
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 & *\\ ...
1
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1answer
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 ...
2
votes
1answer
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 ...
4
votes
2answers
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 ...
6
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1answer
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 ...
1
vote
1answer
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^...
8
votes
1answer
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 ...
2
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1answer
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 ...
2
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1answer
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....
5
votes
1answer
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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0answers
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 ...
2
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0answers
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 ...
4
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1answer
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
votes
1answer
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 ...