3
votes
0answers
113 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 ...
7
votes
2answers
432 views

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 ...
3
votes
1answer
236 views

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 $F'$ ...
0
votes
0answers
100 views

Hilbert Class fields and Pure cubic fields

I want to know about Hilbert class field theory for pure cubic fields. Which is the best source for this? Right now I am reading book by D.A. Cox "Primes of the form $x^2+ny^2$" Fermat,Class field ...
10
votes
0answers
180 views

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 ...
3
votes
1answer
177 views

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 ...
10
votes
3answers
443 views

Ideal classes fixed by the Galois group

Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...
25
votes
0answers
439 views

Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
2
votes
0answers
122 views

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 $\psi_p$, ...
1
vote
0answers
154 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 ...
6
votes
1answer
315 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 ...
7
votes
1answer
377 views

Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...
2
votes
0answers
160 views

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 ...
6
votes
0answers
178 views

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 ...
3
votes
1answer
231 views

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 ...
5
votes
0answers
255 views

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 ...
4
votes
2answers
328 views

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 ...
4
votes
1answer
365 views

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 ...
4
votes
2answers
286 views

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. ...
1
vote
2answers
296 views

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). ...
1
vote
0answers
240 views

Ray class field and ring class field

Let $K$ be quadratic number field and let $O$ be an order of $K$. A modulus $m$ of $K$ is a formal product $m_0\cdot m_\infty$ of finitely many finite primes $m_0$ and finitely many infinite real ...
3
votes
1answer
311 views

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 ...
6
votes
2answers
368 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 ...
7
votes
1answer
464 views

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 ...
7
votes
1answer
549 views

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 ...
3
votes
0answers
491 views

Characterizing primes that split completely vs. primes with a given splitting behavior

Given a finite abelian extension of number fields $L/K$, the prime ideals $\mathfrak{p}$ in $O_K$ split into primes $\mathfrak{P}$ in $O_L$. The number of primes $\mathfrak{p}$ splits into is ...
3
votes
1answer
399 views

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 ...
4
votes
0answers
251 views

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 $4a^3+27b^2=c^2$ ...
4
votes
2answers
356 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 ...
9
votes
1answer
650 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 ...
12
votes
2answers
907 views

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 ...
9
votes
0answers
611 views

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, ...
14
votes
5answers
985 views

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 ...
9
votes
0answers
581 views

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: ...
20
votes
1answer
926 views

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 ...
3
votes
0answers
216 views

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 ...
11
votes
0answers
365 views

Lubin-Tate vs cohomological approach to local CFT

Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes ...
13
votes
1answer
447 views

Embedding number fields in fields with class number 1

(Apologies if this question isn't quite research-level: a colleague came across it while preparing a non-examinable bonus lecture on class field theory for an undergraduate algebraic number theory ...
8
votes
1answer
497 views

class numbers of $\mathbf{Q}(2^{1/n})$

Calculating the class numbers of $\mathbf{Q}(2^{1/n})$ for small $n$ always yields $1$. Is it true for an infinite number of $n$s? Does applying Iwasawa theory to the false Tate curve tower ...
6
votes
1answer
424 views

Parity of class number of pure cubic fields

A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$. What can be said about the parity (odd or even) of the class number of a pure ...
4
votes
2answers
458 views

How do Brauer groups relate to zeta functions?

There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the ...
16
votes
2answers
1k 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) ...
13
votes
5answers
2k views

What is the “reason” for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? . I ...
10
votes
0answers
493 views

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 $p$ ...
16
votes
1answer
792 views

Class number parity in pure cubic number fields

Consider the family of pure cubic number fields $K = {\mathbb Q}(\sqrt[3]{m})$ for $m = a^3 \pm 3$. Proposition. If $4 \mid a$ and $m$ is cubefree, then the class number of $K$ is even. Proof. Let ...
2
votes
3answers
869 views

number of galois extensions of local fields of fixed degree

Let $K$ be a local field (of characteristic 0) with (finite) residue field of characteristic $l$ and let $p$ be a prime. Considering the cases, whether the $p$-th roots of unity are in $K$ and ...
1
vote
2answers
787 views

Decomposition of Artin L functions

The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of ...
6
votes
1answer
565 views

Recovering Hecke L-series from Artin L-functions

Let $K$ be a number field, $\chi : C_K \to \mathbb{C}^\ast$ a Hecke character (that is, a character of the idèle class group), and $L(\chi,s)$ the corresponding Hecke $L$-series. I wish to understand ...
4
votes
1answer
537 views

What are the roots of unity in abelian extensions of imaginary quadratic fields?

What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know: Is $K(\zeta_n)/K$ an abelian ...
23
votes
2answers
2k views

Why is Class Field Theory the same as Langlands for GL_1?

I've heard many people say that class field theory is the same as the Langlands conjectures for GL_1 (and more specifically, that local Langlands for GL_1 is the same as local class field theory). ...