The algebraic-number-theory tag has no wiki summary.

**4**

votes

**1**answer

1k views

### Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.
As we all know, ...

**7**

votes

**0**answers

246 views

### Do the Adeles Split?

I asked this question about a week ago here http://math.stackexchange.com/questions/288955/splitting-the-exact-sequence-of-the-idele-class-group, but got no answer so I thought I'd aske here and see ...

**2**

votes

**0**answers

230 views

### galois cohomology over finite field

Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$.
Do we have a formula to ...

**5**

votes

**2**answers

394 views

### On Weil's characters of type (A)

In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite ...

**4**

votes

**0**answers

229 views

### What else does the Tate-Nakayama lemma tell us about class field theory?

Doing class field theory from the point of view of class formations, it is my understanding that to get the artin reciprocity map, one does this by inverting an isomorphism which is given by the ...

**6**

votes

**0**answers

246 views

### Differences in tree picture of ${\bf Q}_p$, $\overline{{\bf Q}_p}$, ${\bf C}_p$, $\Omega_p$

I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value ...

**19**

votes

**4**answers

2k views

### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

**19**

votes

**1**answer

514 views

### Décomposition des nombres premiers dans des extensions non abéliennes

Gauß famously determined the cubic character of $2$ in his Disquisitiones : $2$ is a cube modulo a prime number $p\equiv1\mod3$ if and only if $p=x^2+27y^2$ for some $x,y\in\mathbf{Z}$. This implies ...

**22**

votes

**1**answer

924 views

### Is pi = log_a(b) for some integers a, b > 1?

Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...

**1**

vote

**0**answers

187 views

### Extending systems of l-adic representations to other l

I'm asking this not because I have an idea how one might approach it, but because it seems natural and inherently interesting.
Let $K$ be a number field, $G_K$ its absolute Galois group, and ...

**4**

votes

**0**answers

178 views

### Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?

Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...

**7**

votes

**2**answers

760 views

### Elementary proof of Mordell's theorem

My question concerns a comment in Silverman-Tate's book "Rational Points on Elliptic Curves" (second printing). On page 76 they begin their proof of 'lemma 4' to the proof of Mordell's theorem, which ...

**26**

votes

**1**answer

953 views

### What happened to Emmy Noether's *Zukunftsphantasie* ?

Recenly I came across Peter Roquette's article On the history of Artin's $L$-functions and conductors (23 July 2003) in which he talks about some letters from Emil Artin and Emmy Noether to Helmut ...

**3**

votes

**0**answers

138 views

### Modified radical group of a Kummer extension

If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$ --- let's also assume $K/k$ is not generated by $p$-power roots of unity), I ...

**5**

votes

**0**answers

144 views

### On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...

**24**

votes

**0**answers

719 views

### Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$,
and define the Fekete polynomials
$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$
Define
$$ f_N(X) = ...

**3**

votes

**1**answer

275 views

### Langlands Paper on representations of abelian algebraic groups

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his maps is obvious and ...

**1**

vote

**0**answers

255 views

### Applications of the class number formula, etc.

This is a big list of applications of the class number formula and its generalizations. I'll start:
The solution to Gauss's class number problem for imaginary quadratic fields, and more generally ...

**2**

votes

**0**answers

110 views

### Comparing ideal class numbers of different orders

Let $P$ be a monic irreducible integral polynomial. Let $K=\mathbf Q[X]/(P)$ be the associated number field, $\mathcal O$ be its ring of integers and $R$ be the order $\mathbf Z[X]/(P)$.
(In general, ...

**4**

votes

**3**answers

410 views

### Orders of Number Fields

Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order.
$\textbf{Questions:}$
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\cO}{\mathcal{O}}$
...

**6**

votes

**2**answers

967 views

### Eigenvalues of nonnegative integer matrices

Edit
I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post:
What are the possible eigenvalues of nonnegative integer matrices?
Any answer ...

**6**

votes

**2**answers

769 views

### Explicit examples of algebraic Hecke characters with infinite image?

Jerry Shurman has a lovely set of notes explaining the classical definition of Hecke characters, the idelic definition of Hecke characters, their relationship, and the classification of algebraic ...

**8**

votes

**3**answers

634 views

### A natural way of thinking of the definition of an Artin $L$-function?

Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 ...

**4**

votes

**1**answer

300 views

### Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear?

If $K$ is an imaginary quadratic field, then the $\mathbf{Z}_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}_p$-extensions of $K$ in an algebraic closure ...

**6**

votes

**0**answers

139 views

### How to construct Weil numbers in a given CM quartic field?

Let $L$ be a CM field of degree $4$ over the rationals, and let $p$ be a prime number. If $q$ is a power of $p$, I would like to know if it is possible to characterize (in some way) all Weil ${\bf ...

**0**

votes

**0**answers

55 views

### Commuting invariants and duals of C_p vector spaces

Let $K$ be a field complete with respect to some discrete valuation, with perfect residue field of characteristic $p$. Let $\mathbb{C}_p$ be the completion of an algebraic closure of $K$, and set $G_K ...

**7**

votes

**0**answers

1k views

### What can we do to raise awareness of reciprocity laws? [closed]

The study of reciprocity laws is a centerpiece of modern mathematics. Of the last ten Fields Medalists, two of them (Ngô Bảo Châu and Laurent Lafforgue) were awarded Fields Medals for their work on ...

**5**

votes

**1**answer

479 views

### Heegner Points and Binary Quadratic Forms

I've been trying to read Gross' paper on Heegner points on $X_0(N)$ and I am stuck on a few details. The definition he is working with is that a heegner points is a pair $y=(E,E')$, where $E$ and $E'$ ...

**0**

votes

**1**answer

556 views

### A question on Cebotarev's density theorem

Let $K$ be a number field, $d$ a positive integer and $S$ a finite set of places of $K$.
By Cebotarev, there exists a finite set of finite places $T$ disjoint from $S$ such that the conjugacy classes ...

**1**

vote

**1**answer

251 views

### Around a theorem of Kronecker

Hi,
let $k/\mathbb{Q}$ be a number field. Assume that $u$ is an algebraic integer such that all $k$-conjugates have modulus $1$. Is $u$ a root of $1$ ?
If $k=\mathbb{Q}$, the answer is YES (this is ...

**9**

votes

**1**answer

440 views

### Structure of units in a maximal order

Hello,
my question is simple: do we have a "Dirichlet's unit theorem" for the group of units of a maximal order of a central division algebra ?
In other words: let $k$ be a number field, let $D$ be ...

**2**

votes

**1**answer

244 views

### Is being principal a local property?

Let $R$ be a number ring and a Dedekind domain. We have the following result:
For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the ...

**6**

votes

**1**answer

223 views

### Algebraic integers in skew fields

Hi everyone,
let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a ...

**2**

votes

**1**answer

338 views

### is there any bound on the absolute number of algebraic integer in terms of its degree?

If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))?
Clearly we need to have distinct roots of unity otherwise this ...

**14**

votes

**2**answers

622 views

### Motivic generalisation of Neron-Ogg-Shaferevich criterion

Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...

**6**

votes

**2**answers

405 views

### Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields.
Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$.
In general, the Jacobian $J(X)$ probably ...

**3**

votes

**1**answer

210 views

### Extending arithmetic functions (and associated Dirichlet series) to arbitrary rings of integers

Many classical arithmetic functions can be thought of as functions on the set of (non-zero) ideals of $\mathbb{Z}$ rather than as functions on $\mathbb{N}$.
Example: For $n \in \mathbb{N}$ the ...

**2**

votes

**0**answers

242 views

### Elements whose conjugates are of the same absolute value in cyclotomic fields

Let $k$ be an odd rational integer, $p$ a rational prime and $\zeta_p$ a primitive $p$th root of unity. Let $\sigma$ a generator of $Gal(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, i.e., ...

**1**

vote

**1**answer

211 views

### elements of absolute value one in cyclotomic fields

Let $p$ be a rational prime and $\zeta_p$ a primitive $p$th root of unity.
What do we know about the set $\{z\in\mathbb{Q}(\zeta_p):|z|=1\}$?

**0**

votes

**1**answer

402 views

### When does the modulus of a sum of an integer and an algebraic integer equal an integer?

Let say Z is a sum of n-roots of unity and thus an algebraic integer, and D is an rational integer.
If |z+D| is an integer, what can we conclude regarding Z? can we say |Z| is integer?
Another ...

**2**

votes

**0**answers

171 views

### Proving the index formula of the norm residue group without $p$-adic completion

I asked the following question in MSE, but I've got no answer so far.
Let $K$ be an algebraic number field.
Let $A$ be the ring of integers in $K$.
Let $\mathfrak{p}$ be a prime ideal of $K$.
Let ...

**9**

votes

**1**answer

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 ...

**1**

vote

**1**answer

373 views

### Is every countable Dedekind domain the ring of integers of some number field?

Is every countable Dedekind domain the ring of integers of some number field? I tried googling different keywords, but did not find anything. Does anyone know of research in this area?

**8**

votes

**3**answers

769 views

### which algebraic integers in a cyclotomic field give you integer absolute value?

Does anyone know an answer to this question?
Question: In an cyclotomic field which algebraic integers have integer absolute value?
Revision 1: -1
I like to add this to the above question, Let's ...

**16**

votes

**1**answer

700 views

### Cyclotomic polynomials evaluated at roots of unity

Dear MO_World,
I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning ...

**4**

votes

**2**answers

275 views

### Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q

I'm looking for an explanation of the following result:
If D is a maximal order in a definite (i.e. ramified at infinity) quaternion algebra B over $\mathbb{Q}$, and $\phi : ...

**5**

votes

**1**answer

543 views

### Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...

**5**

votes

**2**answers

403 views

### Book on ideal theory in Hurwitz quaternions

Hello,
I am looking for a book that studies the set of Hurwitz quaternions (HQ). In particular, I am interested in a connection between HQ and imaginary quadratic fields (IQF); quaternion orders ...

**8**

votes

**2**answers

664 views

### What is known about first cohomology of the units in a number field?

Let $K/Q$ be a finite Galois extension with Galois group $G$. Let $U\subset K^\times$ be the group of units. I am interested in any available information about $H^1(G,U)$.
Motivation: in the theory ...

**20**

votes

**9**answers

2k views

### What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?

I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...