All Questions
Tagged with class-field-theory reference-request
38 questions
3
votes
1
answer
123
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Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
3
votes
0
answers
91
views
Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
- 785
3
votes
3
answers
388
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On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
3
votes
0
answers
80
views
Local Class field theory and Artin map for the Weil group
I am searching a reference for local class field theory that use the Weil group instead of the absolute Galois group. In particular that the Artin map is an isomorphism between the multiplicative ...
- 367
2
votes
2
answers
433
views
Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
4
votes
0
answers
160
views
Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields
Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...
- 411
3
votes
0
answers
215
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Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
4
votes
0
answers
181
views
The order of the global Galois group
For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
- 863
5
votes
1
answer
440
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 ...
- 835
0
votes
1
answer
679
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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 ...
35
votes
5
answers
9k
views
A reference for geometric class field theory?
The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...
- 805
25
votes
2
answers
4k
views
Primes of the form $x^2+ny^2$ and congruences.
The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
- 26.1k
10
votes
1
answer
707
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 ...
- 7,746
17
votes
6
answers
3k
views
Reference for learning global class field theory using the original analytic proofs?
I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find either does local ...
- 15.4k
4
votes
0
answers
347
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 ...
- 333
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
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
9
votes
0
answers
890
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
8
votes
5
answers
2k
views
Where can I find online copies of class field theory publications by Kronecker, Weber, Chevalley, Hasse, Hilbert, Takagi, etc?
I am writing an undergraduate thesis on local and global class field theory from a classical (i.e., non-cohomological) approach and am hoping to obtain copies of the early groundbreaking publications ...
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
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
6
votes
2
answers
323
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^+$ ...
- 7,746
5
votes
1
answer
898
views
p-adic expansion for elements in algebraic closure of p-adic numbers
In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. My question is if this "conjecture" has been ...
- 596
8
votes
1
answer
1k
views
Generalization of Hilbert 94 and capitulation
Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
- 5,670
4
votes
0
answers
309
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 $...
- 1,384
4
votes
0
answers
190
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 $...
- 11.3k
5
votes
1
answer
516
views
Reference for Local class field theory via witt vectors
I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...
- 71
9
votes
1
answer
1k
views
Class groups of orders
In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group $\mathrm{Cl}...
- 647
5
votes
1
answer
1k
views
Verlagerung made "explicit"
Suppose given a finite extension $L/K$ of number fields. I would like to develop a better intuition for the Verlagerung giving an embedding $Ver : Gal(K^{ab}/K) \to Gal(L^{ab}/L)$.
For example, let $...
- 2,029
6
votes
2
answers
799
views
the global m-th power reciprocity law and Quartic Reciprocity Law
I'm reading Cox "Primes of the form $x^2+ny^2$". And I read a chapter about the global m-th power reciprocity law. Now I'm not able to prove the quartic and cubic reciprocity laws. Where can i find ...
- 61
11
votes
1
answer
2k
views
The Class Number One Problem for Real Quadratic Fields
An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
23
votes
1
answer
4k
views
Chapters 1--4 of the Artin-Tate notes on Class Field Theory
Emil Artin and John Tate held a seminar on class field theory at Princeton University in 1951--1952. Their notes were published in 1967 by Benjamin (New York), but the first four chapters covering (...
- 18.7k
4
votes
0
answers
559
views
Explicit description/calculation of norm group of ideles of characteristic $p$ global field
I posted the same question earlier in stack exchange,
(https://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...
- 111
4
votes
2
answers
658
views
Intersection of Hilbert class fields of imaginary quadratic fields
In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could ...
- 1,905
2
votes
0
answers
100
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 ...
- 87
2
votes
0
answers
241
views
Reference request: Cohomology of Elliptic Curves
Is it true that the group
$$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$
is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility?
...
- 1,805
4
votes
0
answers
332
views
Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.3k
3
votes
0
answers
158
views
Conics over number fields
I am looking for a reference for the following fact.
Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...
- 21.4k