Questions tagged [class-field-theory]
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382 questions
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Does the fundamental group identify group structure on subvarieties of products of curves?
Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:
$$ \pi_1^{ab}(...
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Unramified extensions of a given degree
Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ?
EDIT: If not then under what conditions on $K$,...
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1
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Finite Nontrivial Unramified Towers of Number Fields
Let $F$ be a number field and $L=F^{un}$ its maximal unramified extension. By Class Field Theory, $$Gal(L/F)^{ab}\cong Cl(F).$$ It's well-known that we can have $[L:F]=1$ (e.g. $F=\mathbb{Q}$), and $[...
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Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
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2
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1
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A certain idele class character
Let $E/K$ be a cubic extension of number fields, $\nu$ be a Grossencharacter of the idele class group $\mathbb{I}_{E}/E^{\ast}$ such that $\nu^2$ is trivial and $\nu$ restricted to the idele class ...
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Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)
We have the Lucas numbers, $$ L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; . $$
Question: is it the case that
$$ f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z $$...
CommunityBot
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Textbook request for class field theory [duplicate]
I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...
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Skew symmetry for the Hilbert symbol
Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,...
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Reciprocity laws in different dimensions
Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to −\...
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Relation between 1-dimensional and 2-dimensional reciprocity maps
Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, \...
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1
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1
answer
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Equivalence of definitions of the Milnor $K$-groups
In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as
$(R^{\times}...
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Galois groups and prescribed ramification
What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
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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 ...
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3
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1
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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 $F'$ ...
- 1,285
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570
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Cubic fields correspond to $3$-torsion ideals in quadratic fields, or to order $3$ characters of quadratic class groups?
I was watching Dick Gross's laudation for Manjul Bhargava, followed up by one of Bhargava's talks, and I realized I was confused about something.
Bhargava says (around 21 minutes) that the orbits of ...
- 156k
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1
answer
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Cyclotomic character in class field theory
Let $K$ be an extension of $\mathbb{Q}_p$.
By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}_K \rightarrow \mathbb{Z}_p^\times$ corresponds to a character $\chi : K^\times \...
- 149k
5
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Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4
In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...
CommunityBot
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1
answer
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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}...
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Global Artin reciprocity law from Local class field theory
Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...
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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 ...
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4
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1
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About principal ideal theorem in number fields
I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$.
In this case, there is a well-known result that "every ambiguous class in the class group $\...
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Ring class fields of orders
The theory of ring class fields corresponding to orders in imaginary quadratic fields is introduced by, say, [1]. But I'm reading an article [2], in which ring class fields corresponding to orders in ...
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100
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divisor class group with modulus
Let $C$ be a smooth projective curve over a field $k$ and $S \subset C$ a finite number of points. A modulus is simply a divisor supported on $S$. What is the divisor class group with modulus?
I ...
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n-dimensional local fields
Recently, I hear the concept of $n$-dimensional local fields.
It is defined inductively as follows.
(1) a $0$-dimensional local field is a finite field.
(2) an $n$-dimensional local field is a ...
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Extending methods from Lubin-Tate theory
The first lemma in Lubin-Tate theory says the following:
Let $K$ be a local field, $A$ its ring
of integers, and $f\in A[[T]]$ be such
that $f(0) = 0$, $f'(0)$ is a
uniformizer, and $f$ ...
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3
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1
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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 ...
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maximal abelian extension of quadratic extension of $\mathbb Q_p$
I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,...
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2
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On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$
Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...
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1
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Idelic Artin map
I want to study the idelic Artin map for the, say, 7th cyclotomic field explicitly. In other words I want to see how a particular idele can be constructed which gets sent to the identity element in ...
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4
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1
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class groups of unramified cyclic p-extensions of imaginary quadratic fields
Let $K$ be an imaginary quadratic number field with $p$-Sylow-class group $A(K)$ and $L/K$ be an unramified cyclic extension of $K$ of degree $p$ ($p$ prime). Then I am looking for heuristics on
$...
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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.
...
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1
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2
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2-class group of a quadratic imaginary extension
Let $p\equiv 5 [8]$ be a prime number, and consider $K=\mathbb{Q}(\sqrt{-p})$.
I would like to check that the $2$-Sylow subgroup of the class group $C_K$ has order $2$ (I'm pretty sure it's true).
...
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2
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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 ...
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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 ...
CommunityBot
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3
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2
answers
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?
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1
answer
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Fields whose embeddings into the complex numbers are invariant under complex conjugation
Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...
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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 (...
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Numbers integrally represented by a ternary cubic form
Given integers $a,b,c,$ and cubic form
$$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$
$$ f(a,b,c) =
\det \left( \begin{array}{ccc}
a & b &...
- 79.9k
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Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?
Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.
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1
answer
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Can one prove complex multiplication without assuming CFT?
The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
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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:
$rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$
...
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What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
This is related to my first MO question and Kevin Buzzard's conjecture at
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
In December 2010 my question appeared in the M.A.A. Monthly, ...
CommunityBot
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1
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Non-cyclotomic abelian extensions
Suppose $L|\mathbb{Q}$ is an abelian extension of number fields. Then, all the roots of unity are certainly contained in the maximal abelian extension $L^{ab}$ of $L$. Why is it obvious that if $L \ne ...
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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 $4a^3+27b^2=c^2$ ...
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2
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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 ...
CommunityBot
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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 ...
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What is the "ray" in ray class group?
I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a ...
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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 $...
- 156k
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How does one understand geometric CFT in terms of modularity?
I have recently asked a question in a similar vein:
What makes Geometric CFT easier than CFT?
but I'm afraid I wasn't quite ripe to ask it yet. I have since consulted with the following sources:
http:/...
CommunityBot
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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 ...
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