Questions tagged [class-field-theory]
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382 questions
15
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Is there an elegant algebraic proof of this formula for quadratic field discriminants?
Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "...
- 7,347
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0
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177
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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
3
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0
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151
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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-...
- 81
5
votes
1
answer
898
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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
7
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Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed splitting behaviour of $p$
Let $x\in\{\text{totally ramified, inert, totally split}\}.$
If $p\geq 5$ is a prime, are there infinitely many imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of class number coprime to $p$ so ...
- 1,805
10
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0
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358
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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(\...
- 156k
12
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1
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Parametrizing all cyclic extensions of the rational numbers of degree 5
Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...
- 11.3k
2
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1
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$[J_F : P_FN_{E/F}J_E] = 2$ 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 ...
- 277
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What makes Geometric CFT easier than CFT?
I've been reading:
math.stanford.edu/~conrad/249BPage/handouts/geomcft.pdf
in an attempt to shed some geometric light on class field theory. The last paragraph there reads:
In case the ground field $...
- 5,929
12
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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 ...
- 273
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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:...
- 161
5
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1
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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 ...
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2
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What are the current trends in class field theory?
Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that ...
- 3,062
2
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0
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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 $v\in S$, ...
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9
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1
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Globalizing local field extensions with controlled ramification
Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...
- 6,049
2
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0
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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
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4
answers
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Class Field Theory for Imaginary Quadratic Fields
Let $K$ be a quadratic imaginary field, and $E$ an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of $K$. Let $j$ be its $j$-invariant, and $c$ an integral ideal of $...
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3
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Where does the principal ideal theorem (from CFT) go?
My impression is that one of the celebrated results of class field theory the principal ideal theorem namely that given a number field $K$ and its maximum unramified abelian extension L, every ideal ...
- 7,072
1
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0
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138
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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
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382
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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 ...
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3
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Deciding a quadratic diophantine equation
Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$
I am more interested in seeing if there is a quick way to test for case when ...
- 13.9k
2
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1
answer
463
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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 ...
- 87
4
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0
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236
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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
10
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0
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600
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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--...
- 5,780
2
votes
1
answer
202
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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 ...
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1
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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
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0
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211
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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
4
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1
answer
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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 ...
- 11.3k
5
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1
answer
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Why is Kronecker's Jugendtraum only for abelian extensions?
Why is Kronecker's Jugendtraum only for abelian and not for more general extensions of number fields?
Wikipedia, Hilbert's Twelfth Problem.
user19475
13
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1
answer
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How did Takagi prove Kronecker's Jugendtraum for Q(i)?
In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get ...
- 15.4k
12
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1
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Class field towers
It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...
user6976
1
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0
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63
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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
4
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0
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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
5
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1
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478
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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 ...
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0
answers
251
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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 ...
- 1,013
1
vote
1
answer
340
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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 ...
user120812
4
votes
1
answer
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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 ...
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2
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Class groups in dihedral extensions - some sort of Spiegelungssatz?
Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ...
- 13k
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How does Tate cohomology fit into a derived categories framework?
I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented.
...
15
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1
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Quickest and/or most elementary proof of "principal iff splits completely"?
Let $L$ be the Hilbert class field of a number field $K$, and let $\mathfrak{p}$ be a prime ideal of $K$. Then $\mathfrak{p}$ splits completely in $L$ if and only if $\mathfrak{p}$ is a principal ...
user74319
14
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2
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3k
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Elliptic Curves with CM and Class Field Theory
Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known ...
- 235
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470
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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. ...
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1
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253
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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 & *\\
...
- 7,746
2
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0
answers
58
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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
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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 ...
- 21
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1
answer
362
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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 ...
- 1,521
4
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0
answers
190
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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 $...
- 11.3k
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$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 ...
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3
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1k
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Kummer generator for the Ribet extension
Let $p$ be an odd prime and let $k\in[2,p-3]$ be an even integer such that $p$ divides (the numerator of) the Bernoulli number $B_k$ (the coefficient of $T^k/k!$ in the $T$-expansion of $T/(e^T-1)$). ...
- 18.7k
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0
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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 $...
- 1,283