Questions tagged [class-field-theory]
The class-field-theory tag has no usage guidance.
382 questions
3
votes
1
answer
198
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-...
- 4,418
3
votes
0
answers
166
views
Are class numbers of number fields with prime degree often $1$?
I have taken a look at the class number statistics of the L-functions and Modular Forms Database:
https://www.lmfdb.org/NumberField/stats, table "Distribution by class number".
It appears ...
- 125
3
votes
2
answers
558
views
Dedekind Zeta functions of Biquadratic fields
Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...
- 577
2
votes
0
answers
128
views
How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?
I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.
As Soogil Seo ...
- 99
2
votes
1
answer
154
views
Are the maximal cyclotomic field contained in a number field and its Hilbert class group the same?
Let $K$ be a number field. If $d$ be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$ then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(...
- 743
9
votes
1
answer
827
views
how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?
I want to construct the Hilbert class field of $K=\Bbb Q(\zeta_{23}).$ I have no clue how to construct it except that I know that $[H(K):K]=3$ from Sage. Any references or comments are appreciated.
- 343
2
votes
2
answers
469
views
Relation between the Selmer group and the ideal class group
Let $E/K$ be an elliptic curve defined over the number field $K$. Does exist any relation between the $p$-Selmer groups of $E/K$ and the ideal class group $Cl(K)$ of $K$?
- 437
0
votes
1
answer
192
views
English translation of Hasse's "Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage"
I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...
- 577
1
vote
1
answer
138
views
$0$-th Galois cohomology with topological Milnor K-groups coefficients
In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...
- 479
2
votes
0
answers
293
views
Totally ramified extensions of p-adic fields
Let $\mathbb{Q}_p$ denote the field of p-adic numbers. For a prime number $q$ ($\neq p$), does exist a totally ramified extension $K/\mathbb{Q}_p$ with Galois group isomorphis to $\mathbb{Z}_q \times \...
- 437
6
votes
1
answer
1k
views
Set of quadratic forms that represents all primes
A SPECIFIC CASE:
Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$.
If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...
2
votes
1
answer
184
views
Does K and its Hilbert class field have same conductor?
Let $K$ be an abelian number field and $H(K)$ be the Hilbert class field of $K.$
Definition: (conductor of a abelian number field) Let $K$ be a number field with the abelian Galois group over $\Bbb{Q}....
- 343
4
votes
1
answer
580
views
Etale cohomology and Kummer theory
If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
- 63.1k
0
votes
0
answers
140
views
Field of algebraic functions
We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...
- 1
2
votes
0
answers
327
views
Definition of Euler system of cyclotomic units
I am not sure about my understanding of Euler system of cyclotomic unit. This is what I have learnt:
Let $F=\mathbb{Q}(\mu_m)$.
Let $\mathcal{I}(m)$ = {positive square free integers divisible only by ...
- 99
2
votes
1
answer
168
views
What are conditions to satisfied by rational prime p so that every prime lying above p is a prime of order 1 and generates class group?
I was reading a paper on Euclidean ideals by H Graves and M. Ram Murthy. I have a problem in understanding one of the claims.
setup
Let $K$ be a number field and $H(K)$ is its Hilbert class field. ...
- 743
8
votes
1
answer
552
views
Artin reciprocity via Shimura varieties
The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a ...
- 4,418
4
votes
0
answers
506
views
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 ...
- 99
0
votes
0
answers
124
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 ...
- 1,215
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}\...
- 15.4k
4
votes
1
answer
304
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 ...
- 479
0
votes
0
answers
87
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 ...
- 437
12
votes
2
answers
605
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 ...
- 9,501
1
vote
0
answers
2k
views
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=...
1
vote
1
answer
252
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 ...
- 5,966
6
votes
1
answer
451
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 $...
- 743
2
votes
0
answers
168
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$ ...
- 6,180
2
votes
0
answers
119
views
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 ...
- 6,180
2
votes
2
answers
415
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 ...
- 3,362
3
votes
1
answer
386
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$?
- 653
4
votes
1
answer
418
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:
$...
- 5,966
2
votes
1
answer
216
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{...
- 312
2
votes
0
answers
181
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 $...
- 281
3
votes
2
answers
500
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$ (...
user145520
2
votes
1
answer
599
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 ...
- 333
7
votes
1
answer
301
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 ...
- 1,283
-1
votes
2
answers
312
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 ...
- 101
2
votes
0
answers
131
views
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]).$$
It is not hard to prove that
$...
- 2,375
1
vote
0
answers
63
views
The completion of a ray class field
I'm reading some papers doing computations on global class field theory.
And the class field theory in those papers is ideal-theoretic.
Here is a question.
Given a base field $k$ and a modulus(cycle)...
- 1,013
9
votes
0
answers
888
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
3
votes
0
answers
164
views
Using the Hilbert symbol to find nice field extensions
Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
- 2,516
2
votes
0
answers
58
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 ...
- 390
3
votes
0
answers
151
views
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-...
- 81
2
votes
0
answers
117
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 ...
- 87
3
votes
0
answers
177
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 ...
- 2,375
1
vote
0
answers
138
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 $...
- 3,107
2
votes
0
answers
169
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 $v\in S$, ...
- 323
3
votes
0
answers
37
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 $K \subset E \subset F$...
- 151
4
votes
1
answer
399
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 ...
- 1,322
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 ...
- 4,279