Questions tagged [class-field-theory]
The class-field-theory tag has no usage guidance.
382 questions
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Explicit invariant map in local class field theory
Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map:
$$
H^2(\...
- 953
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347
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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 ...
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1
answer
580
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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
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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 ...
- 99
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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. ...
- 1,606
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465
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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 ...
- 647
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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 ...
- 2,375
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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
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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
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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^\...
- 361
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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 ...
- 87
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308
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Complex multiplication and ray class fields
This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
- 41
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extending $p$-adic character of the local intertia to the absolute Galois group
Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism
$$
\chi \colon O_{F_v}^\times \to E^\times.
$$
where $O_{F_v}$...
- 1,190
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301
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Can we prove the uniqueness of the local Artin map by using mostly global class field theory?
Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...
- 6,180
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Is $K^{ur} K^{\pi} = L$?
Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...
- 6,180
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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
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332
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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
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224
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Reciprocity Map and Cycle Class Map
This might be a very naive question but here it goes. Let X be a smooth variety of dimension d over a p-adic field. We have the n part of the rerciprocity map:
$rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$
...
- 235
4
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401
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Abelian cubic extensions of Q[i],
Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...
- 818
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370
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Formal non-CM in local fields
An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the ...
- 15.4k
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3
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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
$$U_k=\{...
- 87
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1
answer
402
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Is there a non-perfect field in which polynomials of large degree are reducible?
It is well-known that, in a real-closed field $K$, every polynomial of degree $>2$ is reducible in $K$. But in this case the characteristic of $K$ is zero.
My question is: there exists a non-...
- 41
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2
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558
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Dedekind Zeta functions of Biquadratic fields
Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...
- 577
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3
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502
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Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected?
I was reading this question The connected component of the idele class group but I am very confused about the structure of the solenoids $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\...
- 547
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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 $K$?
- 437
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651
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Galois cohomology of the field of Laurent series
Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
- 3,472
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395
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ray class field of rational function field
Let $f \in \mathbf{F}_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}_q[T]/(f))^\times$ can be constructed by adjoining torsion points of a Carlitz module....
user19475
3
votes
1
answer
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Knot group of a field extension
Notation:
$L/K$, finite extension of global fields
$K^\times$, unit group of $K$
$L^\times$, units group of $L$
$\mathbb{A}_L^\times$, ideles of $L$
$N_{L/K}$, the norm map
The knot group of an ...
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Lang's proof concerning ray class fields of imaginary quadratic number fields
Crosspost from Math.SE as I did not receive an answer there:
In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $...
- 199
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1
answer
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Theorem 7b of Serre's "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques"
Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...
- 31
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Where am I going wrong in this interpretation of 1-dimensional geometric class field theory?
I posted this on MSE, but didn't get any responses, so I'm reposting here. I tried to write down an example of the main theorem of geometric class field theory, but I must be misunderstanding ...
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1
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Connected-étale sequence for ordinary CM elliptic curves
Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.
Suppose that $p$ is ...
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Textbook request for class field theory [duplicate]
I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...
3
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1
answer
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Unramified extension and class field theory
I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me.
Let $F$ be a ...
- 601
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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
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1
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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
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1
answer
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Projective coordinates over a non UFD ring
Is it true that when the integers of a number field are not a UFD then not every point in projective $n$-space over that field can be given by relatively prime algebraic integer coordinates?
When a ...
- 11.2k
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Local Tate duality for F-vector space
A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...
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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
3
votes
1
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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 ...
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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 ...
3
votes
1
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386
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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
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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
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80
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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
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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
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141
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Cohomology of local fields in positive characteristic
It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
- 41
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162
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relating class number and narrow class number of a real field
I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...
- 637
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0
answers
117
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Finiteness for Galois cohomology for $\mathbb{Z}_p$-module coefficients
I am looking for a general survey on the finite generation properties of
$$H^i(F,\mathbb{Z}_p(j))$$
for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is ...
- 51
3
votes
0
answers
187
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Simplification of links between idele class group and étale cohomology
I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
3
votes
0
answers
224
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Central division algebras over $ \mathbb{Q} $
Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...
- 923