43
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 ...

20
votes

Accepted

### Why are we defining character groups differently for additive and multiplicative group in Tate's thesis?

I think the basic reason for the apparently different definitions boils down to the different topologies of $k^{*}$ versus $k^{+}$.
For simplicity of discussion, let's consider the case that $k= \...

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(\...

19
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 ...

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 ...

16
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 ...

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)...

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 $...

14
votes

### Why are we defining character groups differently for additive and multiplicative group in Tate's thesis?

There's a lot to absorb in Tate's thesis, but it is worth the effort. A quick, concrete answer to your question is that is allows, for $s\in\mathbb C$, functions like $n\to n^{-s}$ to be a ...

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

### 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 ...

11
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 ...

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}...

11
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, $...

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

### 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 ...

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 ...

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 ...

9
votes

### Surjectivity of the norm of units in Galois extensions ramified exactly at one finite prime

A theorem due to Arnold Scholz says that if $K$ has odd class number and $L/K$ is a quadratic extension with a single ramified prime, then the norm map on units is onto.
In general, this does not hold....

9
votes

Accepted

### Fields in which $ -1 $ can't be written as sum of two square elements

In the notation of Lam's Quadratic forms over fields, the Stufe (a German word) or level (its English translation) $s(F)$ of a field is the minimal $n$ such that $-1$ is the sum of $n$ squares.
A ...

8
votes

### Why is Kronecker's Jugendtraum only for abelian extensions?

The bottom line is that in order to have a "Jugendtraum" for a number field $K$, you first want to have a complete class field theory for it.
Kronecker didn't have a general CFT, so his conjecture ...

8
votes

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

Although @znt has gotten the answer through pari, I think it may be instructive to outline my argument.
It all depends on the transition function of Higher Ramification Theory: for a finite extension ...

8
votes

Accepted

### The Genus field and Hilbert class field

Let $K$ be a quadratic field with class group $\mathrm{Cl}(K)$ such that $\mathrm{Cl}(K)\neq \mathrm{Cl}(K)[2]$. Since the Galois group of the genus field is isomorphic to $\mathrm{Cl}(K)[2]$ and the ...

7
votes

Accepted

### p-adic expansion for elements in algebraic closure of p-adic numbers

The following article discusses p-adic expansions in $\overline{\mathbb{Q}}_p$
and of $\mathbb{C}_p$.
Algebraic $p$-adic expansions,
David Lampert,
Journal of Number Theory 23 (1986), 279–284.

7
votes

Accepted

### Analogue of j-invariant for CM fields

The simplest generalisation to abelian surfaces is, I believe, the statement (which is a theorem, not a conjecture) that the Igusa invariants of an abelian surface with CM by $K$ generate an abelian ...

7
votes

### Class field theory - a "dead end"?

The statements ascribed to Rapoport are nonsense --- they must have been garbled in the transmission. I'd guess he may have said that the approaches to nonabelian class field theory before Langlands ...

7
votes

Accepted

### A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

You seem to be expecting that mod $p$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $GL_2(\mathbf{Q}_p)$ and $WD(\...

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