Questions tagged [class-field-theory]

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Euler Systems and Coleman’s Conjecture

I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
Ash's user avatar
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118 views

When is the natural map of Tate cohomology an isomorphism?

First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
Adittya Chaudhuri's user avatar
2 votes
0 answers
75 views

Maximal order of $x^n-d$ and its dependence on $d$

It's well known that the structure of the maximal order of $\mathbb{Q}[\sqrt{d}]$ depends on $d$ modulo $4$: (assuming $d$ is squarefree), the maximal order is $\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\...
David Corwin's user avatar
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4 votes
1 answer
276 views

Topological structure on higher dimensional local fields

Let $F$ be a $n$-dimensional local field. If $n=0$ or $1$, the topological structure on $F$ was well-known, however if $n>1$ i.e, $F$ is a higher dimensional local field, I don't know something ...
M masa's user avatar
  • 479
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0 answers
72 views

Image of extension ideal classes homomorphism in ideal class group under Artin map in class field theory

Let $K/P$ be a finite extension of number fields and $\epsilon_{K/P}:[\mathfrak{a}] \in Cl(P) \rightarrow [\mathfrak{a}.\mathcal{O}_K]\in Cl(K)$ be the ideal class transfer homomorphism. It's well ...
A. Maarefparvar's user avatar
12 votes
2 answers
577 views

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$. Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
LeechLattice's user avatar
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Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$

Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...
Davood Khajehpour's user avatar
1 vote
1 answer
202 views

Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of non-Archimedean local fields where I can find proofs of the following claims about finite extensions $L/K$ of non-Archimedean local ...
user267839's user avatar
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6 votes
1 answer
374 views

How is class of composition of two quadratic fields is related class numbers of quadratic field?

Let $K_1=\Bbb Q(\sqrt{d_1})$ , $K_2=\Bbb Q(\sqrt{d_2})$ and $K=\Bbb Q(\sqrt{d_1},\sqrt{d_2})$.Suppose $h_1,h_2,h$ be class number of $K_1,K_2,K$ respectively. (i) Can we express $h$ in terms of $...
SUNIL PASUPULATI's user avatar
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0 answers
153 views

Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?

In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
D_S's user avatar
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The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
D_S's user avatar
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2 votes
2 answers
365 views

Computing the class group of a quadratic function field

I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...
Lior Bary-Soroker's user avatar
3 votes
1 answer
330 views

Hilbert class field tower

Let $K$ is a number field,and $H_{K}^{i},i=1,2,\cdots$ be its Hilbert class field tower,suppose it is finite,and let $L=H_{K}^{n}$ is the top of the tower. Must $L$ be galois over $K$?
fool rabbit's user avatar
4 votes
1 answer
352 views

Henselian valued fields for characteristic $0$: a characterization

Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property: $...
user267839's user avatar
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2 votes
1 answer
202 views

class group size of cyclotomic field subextension

In the following, let $\mathbb{Q_1}$ denote the subfield of degree $p$ over $\mathbb{Q}$ in the $p^2$- cyclotomic extension. What is the best known upper bound for the size of its class group, $\text{...
SARTHAK GUPTA's user avatar
2 votes
0 answers
148 views

Anticyclotomic extensions via ideles

Let $ K $ be an imaginary quadratic field with ring of integers $ \mathcal{O} $. Let $ \mathcal{O}_{n} = \mathbb{Z} + n \mathcal{O} $ be the order of conductor $ n $. There is an associated extension $...
Chern's user avatar
  • 281
3 votes
2 answers
472 views

The kernel of the global class field theory homomorphism

Let $K$ be a finite extension of $\mathbb{Q}$. Then there is a surjective homomorphism $\theta:C_K\to G_K^{ab}$ from the idele class group to the abelianization of the absolute Galois group of $K$ (...
user avatar
2 votes
1 answer
552 views

A type of principal ideal theorem of class field theory for ramified primes

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. Also let $p$ be a prime number, $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ and $\zeta_{m}$ be a primitive mth root of ...
Ehsan Shahoseini's user avatar
7 votes
1 answer
274 views

$p$-torsion of class groups

Let $p$ be a fixed odd prime and $\ell$ be another prime such that $\ell \equiv 1 \pmod{p}$. Consider the number field $\mathbb{Q}(\zeta_p)$ and its extension $\mathbb{Q}(\zeta_p, \zeta_\ell)$. Note ...
debanjana's user avatar
  • 1,161
-1 votes
2 answers
290 views

On the determination of ambiguous ideal class of the extension $\mathbb{Q}(\zeta_5,\sqrt[5]{m})/\mathbb{Q}(\zeta_5))$

let $L=\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ and $K=\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic fields, we now that $[L:K] = 5$ and $ GAl(L/K) =\langle\sigma\rangle$ so we call $\mathcal{A}$ an ambigous ...
Fouad El's user avatar
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0 answers
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The field generated by the torsion points of an elliptic curve

Let $E$ be an elliptic curve with complex multiplication by an order $\mathcal O$ in an imaginary quadratic field $K$. Let $H=K(j(E))$ and $$L_N=K(j(E),E[N])=H(E[N]).$$ It is not hard to prove that $...
Shimrod's user avatar
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1 vote
0 answers
62 views

The completion of a ray class field

I'm reading some papers doing computations on global class field theory. And the class field theory in those papers is ideal-theoretic. Here is a question. Given a base field $k$ and a modulus(cycle)...
gualterio's user avatar
  • 1,043
9 votes
0 answers
788 views

How many ways are there to teach class field theory?

I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now. I definitely ...
Mikhail Bondarko's user avatar
3 votes
0 answers
153 views

Using the Hilbert symbol to find nice field extensions

Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
Spencer Leslie's user avatar
2 votes
0 answers
55 views

generator of ring class field extension

everyone! I have another questions. Let $K=\mathbb{Q}(\sqrt{-3})$ be an imaginary quadratic field and let $p\equiv 8\mod 9$ be a prime. Denote $H_{3p}$ and $H_{p}$ for the ring class field of $K$ with ...
yhb's user avatar
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0 answers
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Values of Grössencharacter attached to CM elliptic curve

I am trying a cross-post here, as my previous post on stackexchange was not as fruitful as I hoped. The link to the older post is: https://math.stackexchange.com/questions/3327269/values-of-...
Jupp's user avatar
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2 votes
0 answers
112 views

Definition of a Lubin Tate group

Let $L$ be a $p$-adic number field, $\mathcal{O}_L$ its ring of integers, $\pi$ a uniformizer of $L$ and $q$ the cardinality of its residue field. Let $\varphi(t)\in \mathcal{O}_L[[t]]$ be a ...
user223794's user avatar
3 votes
0 answers
176 views

Abelianess of $K(j(E))$

Let $E$ be an elliptic curve with CM by an order in the imaginary quadratic field $K$. Is there some easy way how to prove that the extension $K(j(E))/\mathbb Q$ is abelian? Update In general, the ...
Shimrod's user avatar
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1 vote
0 answers
134 views

Terminology about ramification

Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $...
efs's user avatar
  • 3,099
2 votes
0 answers
159 views

A version of weak approximation with S-integers

Let $k$ be a finite field. Let $K$ a finite extension of $k(t)$. Let $S$ be a finite set of places of $K$. Let $$K_S = \prod_{v\in S} K_v$$ where $K_v$ is the completion of $K$ at $v$. For $v\in S$, ...
Will Dukeminier's user avatar
3 votes
0 answers
36 views

augmentation ideal and $\rho$-isotopic spaces

These are two questions regarding Rubin`s paper - Global units and Ideal class groups in Inventiones 1987. The two questions are from section 3. Let $p$ be a rational prime and $K \subset E \subset F$...
pedro's user avatar
  • 141
4 votes
1 answer
332 views

Dihedral extension of $\mathbb Q$ with small discriminant

Let $K$ be a fixed quadratic number field, say $K=\mathbb Q(\sqrt 5)$. For any integer $n \geq 3,$ I would like to build a number field $D_n$ such that $D_n/\mathbb Q$ is Galois, with Galois group ...
A. Bailleul's user avatar
  • 1,164
13 votes
1 answer
1k views

Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following: Lemma: if L/K is an abelian ...
Alison Miller's user avatar
63 votes
3 answers
4k views

Class field theory - a "dead end"?

I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes ...
wood's user avatar
  • 2,714
21 votes
0 answers
764 views

Class field theory and the class group

Let $k$ be a finite abelian extension of $\mathbb{Q}$. Class field theory states that $k$ corresponds to some open subgroup of finite index $U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ where $...
Daniel Loughran's user avatar
4 votes
0 answers
193 views

Are there "elementary" proofs of the openness of norm subgroups and of the norm limitation theorem?

Let $K$ be a local field and $L/K$ be a finite extension. Let $L^{ab}$ be the maximal abelian subextension of $K$ in $L$. Write $N_L$ (resp. $N_{L^{ab}}$) for the image of the norm map from $L$ (resp. ...
Simon Pepin Lehalleur's user avatar
4 votes
0 answers
447 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 ...
wkf's user avatar
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12 votes
0 answers
257 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 ...
Estus's user avatar
  • 273
2 votes
0 answers
104 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 ...
quas93's user avatar
  • 21
10 votes
0 answers
333 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(\...
David E Speyer's user avatar
4 votes
0 answers
233 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 ...
Shimrod's user avatar
  • 2,335
8 votes
1 answer
507 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 ...
pierre de fermat's user avatar
1 vote
0 answers
52 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 $...
Pablo's user avatar
  • 11.2k
5 votes
0 answers
120 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 $...
debanjana's user avatar
  • 1,161
6 votes
0 answers
138 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 ...
王李远's user avatar
  • 343
3 votes
0 answers
105 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 $\...
debanjana's user avatar
  • 1,161
2 votes
1 answer
201 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 ...
Pierre21's user avatar
  • 385
9 votes
1 answer
534 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 ...
Jz Pan's user avatar
  • 153
11 votes
0 answers
359 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 ...
Vesselin Dimitrov's user avatar
3 votes
0 answers
242 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 ...
gualterio's user avatar
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