81 votes
Accepted

Motivating Lubin-Tate theory

Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening. Since I had read and enjoyed Lazard’s paper on ...
  • 3,828
40 votes
Accepted

Any open Langlands Conjectures for GL_1?

The question is subject to interpretation, because the Langlands program was never a clearly delimited set of conjectures to begin with, and moreover many things were added during the following ...
  • 25k
25 votes
Accepted

Degree 17 number fields ramified only at 2

Thank you for calling this problem to my attention. I computed $K$ en route to AWS (though this year's topics are a rather different flavor of number theory...). After some simplification (gp's $\rm ...
19 votes
Accepted

Artin reciprocity $\implies $ Cubic reciprocity

I was just thinking about this the other day! Here is one solution. As in the original post, let $K = \mathbb{Q}(\sqrt{-3})$ and let $\theta$ and $\pi$ be distinct primary primes. Also, set $L = K(\...
18 votes
Accepted

how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?

The cyclotomic field $K=\mathbb{Q}(\zeta_{23})$ contains the quadratic field $F=\mathbb{Q}(\sqrt{-23})$, and $F$ has class number $3$ (it is the first quadratic field, when those are ordered by ...
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17 votes
Accepted

On the history of the Artin Reciprocity Law

As GH has already remarked, the same thing happened a lot later after Taniyama and Shimura asked whether elliptic curves defined over the rationals are modular. To begin with your last question, there ...
17 votes

Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the ...
15 votes
Accepted

Galois groups and prescribed ramification

Let me first note that there is a slight ambiguity when one says "ramified only at 2". Strictly speaking, that means that the extension is unramified at every place of $\mathbb Q$ except 2, including ...
  • 25k
14 votes
Accepted

How to compute with the Stark conjectures?

The main reference here is the very useful User's Guide, C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, "User's Guide to PARI / GP" (2003) Particularly the sections about bnrstark (pp. 108)...
  • 17.1k
14 votes
Accepted

Set of quadratic forms that represents all primes

Every prime $p$ is represented by at least one of the following quadratic forms: $x^2+y^2$, $x^2+3y^2$, $3x^2-y^2$: if $p=2$ or $p\equiv 1\pmod{4}$, then $p$ is represented by $x^2+y^2$; if $p=3$ or $...
13 votes
Accepted

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

We give a uniform approach to $p \leq 61$ by applying analytic discriminant bounds to the Hilbert class field. To be sure this is not entirely "conceptual", but then some computation is needed even to ...
12 votes

number of galois extensions of local fields of fixed degree

I am sorry if I see this question only now, but since no one gave the following answer, it seems worth posting it. There is a general formula for the number of extensions of degree $d$ of a $p$-adic ...
12 votes
Accepted

Finite Galois module whose Ш¹ is nonzero?

Wang's conterexample to Grunwald's theorem: $K=\mathbb{Q}(\sqrt{7})$ and $M=\mu_8$. Then $H^1(K,M) \cong K^\times/(K^\times)^8$. Now $16$ is not an $8$-th power in this field but locally an $8$-th ...
11 votes

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Here is an elementary proof, the basic idea for which is in Selberg's 1949 paper "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression" (Ann. Math., vol 2, 1949, pp. ...
11 votes
Accepted

Class field towers

Let $\ell$ be an odd prime and $m$ an integer such that $$|\{p|m,\ p\equiv1\operatorname{mod} \ell\}|\geq8.$$ Then Y.Furuta proved that $\mathbb Q(\zeta_m)$ admits an infinite unramified $\ell$-class ...
  • 9,570
11 votes

Motivating Lubin-Tate theory

Abelian extensions of $\mathbb{Q}$ can be described using torsion points in the multiplicative group. If $K$ is a quadratic imaginary field, and $E$ is an elliptic curve where $\mathcal{O}_K$ acts by ...
11 votes

Unramified extensions of quadratic fields

The following result is a special case of a sideresult in my PhD thesis (1995): Let $k = {\mathbb Q}(\sqrt{d})$ be a quadratic extension with discriminant $d$, let $d = d_1d_2d_3d_4$ be a ...
11 votes

Parametrizing all cyclic extensions of the rational numbers of degree 5

Evidently no. In "Generic Polynomials Constructive Aspects of the Inverse Galois Problem" by Jensen, Ledet, Yui (2002): generic dimension (i.e. minimum number of parameters in a generic polynomial) ...
11 votes

What are the primes that are ramified?

All the information on the higher ramification groups can be derived from Theorem II.5.6, which TKe notes. The Galois group of the ray class field is the group of fractional ideals relatively prime ...
  • 119k
11 votes
Accepted

Chebotarev density theorem and pure weight local systems

As Piotr says, we must assume $U$ normal. The purity assumption is not needed. There are two steps to this proof (1) Suppose for any $u \in U$ and $k \in \mathbb{N}$, we have $Tr(\sigma_u^k,\mathcal{L}...
  • 119k
10 votes
Accepted

Ideal classes fixed by the Galois group

I assume you want $K$ to be Galois over $\mathbb{Q}$. More generally, let $L/K$ be a Galois extension of number fields. The the class group $C_K$ of $K$ maps to $C_L^{G_{L/K}}$, the part of $C_L$ ...
10 votes
Accepted

Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed splitting behaviour of $p$

A more general version of this statement was shown by Kimura in Acta Arith. (2003). His corollary gives $\gg \sqrt{X}/\log X$ such fields ${\Bbb Q}(\sqrt{-d})$ with $d\le X$, and also allows you to ...
  • 42.6k
10 votes
Accepted

Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?

As Serre seems to be fond of saying, life begins at $H^3$. The first appearance of a $3$-cocycle seems to be in Teichmüller's article Über die sogenannte nichkommutative Galoissche Theorie... in ...
10 votes
Accepted

How is class of composition of two quadratic fields is related class numbers of quadratic field?

I assume you mean $K_1=\mathbb Q(\sqrt{d_1})$. 1) If by $K$ you mean $\mathbb Q(\sqrt{d_1d_2})$, then there is no simple relation between $h$ and $h_1$ and $h_2$. 2) If by $K$ you mean the quartic ...
  • 9,361
10 votes
Accepted

Artin reciprocity via Shimura varieties

If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times_+ \backslash \mathbf{A}^\times_{\mathrm{f}} / K$ is exactly $\mu_N / \mathbf{Q}$, the scheme of $N$-th roots ...
10 votes
Accepted

Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?

Here is an example. Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C_2 \times C_2$ and its Hilbert class field is $H_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, $...
9 votes

Are the abelian absolute Galois groups of these local fields isomorphic?

This is how to answer the question (but it's not an answer). (Edit: it and the comments below now form an answer). Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're ...
  • 355
9 votes

Class field theory - a "dead end"?

Let me address your questions 1. - 4. What were the original goals of class field theory? The question is a little bit anachronistic; class field theory describes the splitting of primes in abelian ...
8 votes
Accepted

Cyclotomic character in class field theory

If $K/L$ is an extension of fields, then the natural map $\operatorname{Gal}(K)^{ab} \to \operatorname{Gal}(L)^{ab}$ corresponds in class field theory to the norm map $K^\times \to L^\times$. So you ...
  • 119k
8 votes

Galois groups and prescribed ramification

Concerning the question of Pablo that follows Joël's answer: If $k$ is an algebraically closed field, then the situation is completely understood, thanks to work of Grothendieck (in characteristic 0) ...
  • 12.5k

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