Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale ...

learn more… | top users | synonyms

3
votes
0answers
147 views

Fields generated by torsion points of CM elliptic curves

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm ...
2
votes
1answer
91 views

Two questions about arithmetically equivalent number fields

Two algebraic number fields are said to be arithmetically equivalent if they share the same Dedekind zeta function. If this is the case, they must have certain invariants in common among which is the ...
7
votes
0answers
124 views

Efficient Dirichlet approximation (continued fractions?) over a number field

Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...
1
vote
0answers
220 views

$\mathfrak{q}$-ideal class bound

Let $K$ be a number field, $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{q}$ be a nonzero ideal in $\mathcal{O}_K$. The $\mathfrak{q}$-ideal class group consists of equivalence classes of ...
3
votes
1answer
185 views

Is there an infinite family of primes $q_{1},q_{2},…$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. Much less is known if $K$ is infinite-...
5
votes
2answers
307 views

n-th root of unity in n-th division field of abelian variety?

Let $K$ be a number field and $A/K$ an abelian variety over it. Can it be that $K(A[n])$ does not contain a primitive $n$-th rooth of unity? If the answer is yes is it always possible to ...
2
votes
0answers
192 views

Is it clear that $y^3=f(x)$ has bad reduction at $3$?

Bad reduction is defined as 'nonexistence' of a model where the curve has good reduction. So let's take the curve $C$ which is affinely given by $$y^3 = f(x)$$ (absolutely irred, $f$ no multiple roots)...
4
votes
2answers
283 views

Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},…)$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. The picture is less clear if $K$ is ...
0
votes
1answer
215 views

S. Chowla real quadratic fields

Let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic fields. S. Chowla conjectured that if $d=4m^2+1$ for $m \in \mathbb{Z}$ then there is exactly 6 real quadratic fields which has class number one. Hideo ...
6
votes
1answer
206 views

Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting: the minimal polynomial of the field over $\mathbb{Q}$, and a decimal ...
4
votes
1answer
226 views

How to construct an abelian variety with CM by a given CM field?

Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$. Then $K$ is a so called CM field. For instance, take $F = \mathbb{Q}(\...
5
votes
1answer
523 views

On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$

It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...
12
votes
3answers
606 views

2-torsion in class groups of cubic fields

I was wondering if there are good bounds for the $p$-parts of the class group of a number field $F$ in terms of its discriminant $D_F$. More precisely, the bound for the order of the full class group ...
2
votes
0answers
31 views

Embedding of quadratic field orthogonal to a rational element in quaternion algebra

Let $a,b \in \mathbb{Q}$ and $\mathbb{K}$ be a number field. Consider the quaternion algebra $\left(\dfrac{a,b}{\mathbb{Q}}\right)$ and its $\mathbb{K}$-points $\left(\dfrac{a,b}{\mathbb{K}}\right)$. ...
6
votes
2answers
736 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 $Q$...
8
votes
1answer
205 views

Is the image of an $S$-arithmetic subgroup under a surjective $k$-morphism $S$-arithmetic?

Let $k$ be a global field and let $S$ be a non-empty set of places containing all archimedean ones. Suppose $f:G\to H$ is a surjective $k$-morphism of $k$-groups and let $\Gamma\leq G(k)$ be an $S$-...
2
votes
1answer
172 views

Trace of roots of unity has valuation more than 1 in uramified field

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...
1
vote
3answers
818 views

on the set of numbers generated by integer linear combination of two real numbers.

Let $b > a > 0$ be two real numbers. I am interested in the set of numbers $X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$. What ...
2
votes
1answer
344 views

Some identities with the Riemann-Hurwitz zeta function

The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this, For $0 < a \leq 1$ we have the identity $$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...
18
votes
1answer
636 views

On the history of the Artin Reciprocity Law

At the beginning of Milne's notes on class field theory, he has a quote by Emil Artin (as recalled by Mattuck in Recountings: Conversations with MIT mathematicians): I will tell you a story about ...
6
votes
4answers
262 views

Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not co-compact

Is there a natural example of a discrete subgroup $\Gamma\leq PSL_2(\mathbf{R})$ such that (1) $\Gamma$ has finite covolume (2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper ...
5
votes
1answer
174 views

Relation between ramification locus of a tower and of its constant field extension

I am trying to understand Remark 7.2.22 (Page 256) of Algebraic Function Fields and Codes (Second Edition) by Henning Stichtenoth. In that remark he considers a tower $\mathcal{F} = (F_0,F_1,F_2,\...
0
votes
0answers
56 views

Counts of congruent numbers

I am compiling a table of the total number of congruent numbers $\le 10^{n}$ (see OEIS 274403: http://oeis.org/A274403). So far I have evaluated up to $n = 7$. My question is - Are there computer ...
6
votes
2answers
479 views

Reference on ideal theory in Hurwitz quaternions

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 $\mathcal{O}(...
16
votes
1answer
900 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 ...
0
votes
0answers
62 views

Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p

One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor. For ...
2
votes
1answer
143 views

Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ ...
1
vote
0answers
102 views

Combinatorial splitting in number rings

The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring. Take an arbitrary non empty ...
9
votes
2answers
1k views

What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
5
votes
0answers
367 views

Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and $\mathbb{Q}(\...
3
votes
0answers
103 views

Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...
21
votes
4answers
1k views

Hasse principle for rational times square

Does a Hasse principle hold for the property of being a rational times a square ? Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
7
votes
0answers
271 views

Capitulation of ideal classes in general Dedekind Domains

I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up: Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify ...
11
votes
1answer
208 views

A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic? This question is motivated by the ...
27
votes
8answers
8k views

Practical applications of algebraic number theory?

I'm interested in learning about any applications, the more worldly the better*. Pointing to a nice reference on the number field sieve, for example, would be fine. However, let me mention one ...
0
votes
0answers
56 views

Congruent numbers and primorials

The first 10 primorials (2, 6, ... , 6469693230) are congruent numbers subject to the Birch Swinnerton-Dyer conjecture. My question is - What is the first primorial not to be a congruent number (...
0
votes
0answers
38 views

Is that possible to use stieltjes transform for multiple matrices

I have the matrix calculation with expression \begin{equation} \frac{1}{M}tr(\mathbf{WHH}^H\mathbf{W}^H + \mathbf{R}_{nn})^{-1} \end{equation} whereas $\mathbf{H} \in \mathbb{C}^{M\times K}$, $\mathbf{...
8
votes
1answer
226 views

Imaginary quadratic fields: Euclidean if and only if norm Euclidean

Let $K$ be an imaginary quadratic field and $O_K$ be its ring of integers. We say $O_K$ is norm Euclidean if the norm is a Euclidean function. It is known from the classification of imaginary ...
11
votes
2answers
1k views

References for Artin motives

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
2
votes
1answer
226 views

Non-negative integer solutions of x^2+y^3=n

I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant. I want to know the exact number of non-negative integer solutions. Also I want to know what are those solutions. How ...
16
votes
1answer
796 views

Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation: $$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$ If $\rho$ is odd, it was conjectured by Serre in ...
2
votes
3answers
409 views

Learning roadmap for algebraic number theory

I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am ...
18
votes
2answers
1k views

Context for “Coronidis Loco” from Weil's Basic Number Theory

In Samuel James Patterson's article titled Gauss Sums in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Patterson says "Hecke [proved] a beautiful theorem on the different ...
13
votes
3answers
459 views

Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite? This would complete the answer of Daniel Loughran. There is a ...
2
votes
0answers
54 views

Prescribed norm residue symbol in number field

Suppose $F$ is a number field, and $a, b$ are non-zero elements. Does there always exist $x \in F$ such that the norm residue symbols (=cup products) are $(a, x)= 0 = (x, b) \in H^2(F, \mathbb{F}_2)$ ...
14
votes
8answers
6k views

Suggestions for good books on class field theory

Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field ...
0
votes
1answer
228 views

On the quadratic reciprocity law? [closed]

In the Quadratic Reciprocity Law $$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$. Is there any relation between $x$ and $y$ ...
5
votes
0answers
95 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
5
votes
0answers
115 views

Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...
1
vote
2answers
313 views

Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles: Dickson, L. E.. (1917). Fermat's Last ...