The class-field-theory tag has no usage guidance.
6
votes
0answers
112 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
votes
0answers
91 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 $\...
-1
votes
0answers
65 views
Constructive Kronecker-Weber
Following up on a previous question, where should I look for an algorithm to represent elements of abelian extensions of $\mathbb{Q}$ in a cyclotomic extension of it?
In particular, I am looking to ...
2
votes
1answer
152 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
votes
1answer
203 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 ...
8
votes
0answers
153 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 ...
0
votes
0answers
52 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
votes
0answers
190 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
129 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
votes
1answer
113 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
votes
0answers
78 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
votes
0answers
147 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
votes
0answers
180 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
138 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
148 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
votes
0answers
130 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
232 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 ...
9
votes
0answers
371 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
196 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
72 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
535 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
votes
1answer
247 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
votes
0answers
140 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
votes
0answers
92 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 $v$:
$$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...
7
votes
0answers
141 views
Explicit $H^2(K, \mu) = Q/Z$?
In the development of local class field theory, a very fundamental theorem is that, for every local field $K$ of characteristic zero,
$H^2(K, \mu) \cong \mathbb{Q}/\mathbb{Z}$. $(*)$
Neukirch et al. ...
3
votes
0answers
93 views
Uniqueness of class field theory map
Let $F$ be a local field of characteristic 0. The main theorems in local class field theory can be summarized by the existence of a group $W_F$ and a map
$$
\phi_F:W_F\to W_F^\mathrm{ab}\simeq F^\...
0
votes
0answers
152 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\...
0
votes
0answers
176 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 $\alpha,\...
4
votes
1answer
177 views
Examples of norm forms where the numbers represented can be readily described
In case of impatience: the question here is a request for examples, especially degree six or seven where the norm form might represent some prime$p,$ then some $q^2$ but not $q,$ then some $r^3$ but ...
8
votes
1answer
166 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
vote
1answer
104 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
187 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 ...
3
votes
2answers
210 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
votes
1answer
202 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
150 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
631 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
votes
1answer
194 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
votes
1answer
159 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
203 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 ...
4
votes
0answers
178 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
votes
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
votes
1answer
301 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
228 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 ...
2
votes
0answers
210 views
Valuation topology vs modified valuation topology
Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...
4
votes
1answer
427 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 ...
4
votes
1answer
208 views
Ray class groups through binary quadratic forms
(Cross-posted from https://math.stackexchange.com/questions/2029407/ray-class-groups-through-binary-quadratic-forms)
If $d$ is the discriminant of a quadratic number field, then the primitive classes ...
36
votes
0answers
1k views
A three-line proof of global class field theory?
There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern ...
4
votes
0answers
166 views
Real field of definition of an abelian variety of CM-type?
Question 0. Can a field of definitions (without automorphisms) of an (almost arbitrary) abelian variety of CM-type, originally defined over ${\mathbb{C}}$,
be chosen to be a totally real number ...
22
votes
1answer
1k views
Any open Langlands Conjectures for GL_1?
Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL_n$ which are still open for $n=1$?
7
votes
4answers
474 views
Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?
I am reading Class field theory - Bonn Lectures by Neukirch.
Given a $G$ module $A$ he defines Cohomology groups $H^i(G,A) : i\in \mathbb{Z}$ by considering some complete resolution of $G$ modules ...
