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8
votes
3answers
631 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
1answer
274 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
0answers
138 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
0answers
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
0answers
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
1answer
473 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
1answer
552 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
1answer
249 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
1answer
424 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
1answer
241 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
1answer
222 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
1answer
337 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
2answers
610 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
2answers
402 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
1answer
207 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
0answers
241 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
1answer
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
1answer
400 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
0answers
169 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
1answer
636 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
1answer
369 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
3answers
760 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
1answer
686 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
2answers
272 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
1answer
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
2answers
395 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
2answers
651 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 ...
19
votes
9answers
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 ...
13
votes
2answers
686 views

Did Hermite really prove “Hermite's Theorem” on number field discriminants?

Hermite's theorem, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant. The usual proof (see Neukirch's Algebraic Number Theory ...
2
votes
2answers
337 views

Unsolved problem related Gauss sum and root of unity.

Is there any unsolved problem related Gauss sum or more generally some kind of a sum of roots of unity? Also I would like to know if there is an unsolved problem that can be proved if some (unproved) ...
2
votes
2answers
239 views

Are there formulas for the derivatives $\zeta_{F}^{(n)}(0)$ of Dedekind zeta functions?

Let $F/\mathbb{Q}$ be a number field. I'm interested in knowing if there are formulas for the values of the derivatives $\zeta_{F}^{(n)}(0)$ of the Dedekind zeta function of $F$ at zero. Maybe if in ...
5
votes
2answers
560 views

Explicit description of a quaternion algebra with a prescribed set of ramified places

Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that ...
4
votes
0answers
250 views

On Stickelberger's Theorem over function fields

Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields). Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...
0
votes
2answers
1k views

Fibonacci Numbers Modulo m [closed]

In the paper "Fibonacci Series Modulo m" by D.D. Wall (found here), there is a table in the Appendix listing values for the function $k(p)$. This function is defined as the period of the Fibonacci ...
0
votes
0answers
183 views

Irreducible polynomial?

Let $K$ be the splitting field of a polynomial $f\in \mathbb Q[x]$, which is irreducible mod 3, with $G:=Gal(K|\mathbb Q)=S_n$ (symmetric group). Let $U$ be a subgroup of $G$ with fixed field ...
0
votes
0answers
179 views

Sharpenings of Liouville's inequality

The norm of an algebraic number $\alpha$ is the product of its conjugates, $N(\alpha)$. Suppose that I have an inequality of the form $|x-\alpha*y| > c X^{n-\gamma}$ where $X=max{|x|,|y|}$ and c ...
5
votes
0answers
255 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...
4
votes
1answer
379 views

Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'

I was reading Milne's book "Arithmetic Duality Theorems". On page 166 there are a lot of useful lemmas on the etale cohomology with compact support on S-integers. However, I get confused when I tried ...
3
votes
0answers
412 views

A proof of a theorem on the different in algebraic number fields

I asked this question in Stack Exchange, here, but I got no answer so far. I don't know any modern book on algebraic number theory which states the following theorem, let alone its proof except ...
5
votes
1answer
410 views

Is $\mathbb{Z}[2\cos(\frac{\pi}{k})]$ a Euclidean domain?

The ring $\mathbb{Z}[2\cos(\frac{\pi}{k})]$ is known to be a Euclidean domain for $k=3,4,5$ and $6$, because in those cases $2\cos(\frac{\pi}{k}) = 1, \sqrt{2},$ the golden ratio $\phi$, and ...
2
votes
1answer
432 views

Conductor of an elliptic curve

Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves, there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...
9
votes
1answer
355 views

Which formulae of Euler is Fröhlich referring to?

In A. Fröhlich's article Local Fields in Algebraic Number Theory, the following claim is made: if $R$ is a Dedekind domain with field of fractions $K$, $L$ is a finite separable extension of $K$ and ...
3
votes
1answer
345 views

Proof of a Simple Converse in Algebraic Number Theory

If $L/K$ is a Galois extension, then any prime $\mathfrak{p}$ of $K$ splits into a product ${\mathfrak P}_1^e\cdots {\mathfrak P}_g^e$ of primes in $L$, and the exponents on the primes are equal since ...
7
votes
1answer
2k views

Order of Ш (Sha)

To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not ...
4
votes
2answers
2k views

Ideal theoretic proof of the second inequality of global class field theory

I posted this question in Stack Exchange, but got no answer nor positive vote. So I crosspost this here. Classically the second(or the first in the old terminology) inequality of global class field ...
16
votes
3answers
1k views

Commutative Algebra with a View Toward Algebraic Number Theory

Someone asked me this today, and I don't know what the standard answer is: Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...
6
votes
1answer
375 views

Generalization of Hilbert 94 and capitulation

Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
8
votes
3answers
547 views

j-invariant fixed point?

If we view the j-invariant of a lattice as a map from the upper-half plane to the complexes by $\tau\mapsto j([1,\tau])$, then it is surjective, holomorphic, and has quite a number of other wonderful ...
7
votes
2answers
633 views

Are Cantor type numbers algebraic?

This question may be naive. Take an infinite set of distinct algebraic numbers (hence countable). List them out in a table (randomly) by picking a choice of ordering and change the diagonal numbers. ...
12
votes
1answer
442 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 ...