**3**

votes

**0**answers

150 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

**1**answer

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 ...

**2**

votes

**0**answers

193 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)...

**7**

votes

**0**answers

133 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 ...

**3**

votes

**1**answer

186 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-...

**4**

votes

**2**answers

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

**1**answer

216 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 ...

**4**

votes

**1**answer

227 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}(\...

**12**

votes

**3**answers

609 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

**0**answers

32 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 $\left(\dfrac{a,b}{\mathbb{K}}\right)$. ...

**8**

votes

**1**answer

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

**1**answer

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 ...

**18**

votes

**1**answer

639 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 ...

**0**

votes

**0**answers

57 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

**4**answers

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 ...

**0**

votes

**0**answers

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

**1**answer

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

**0**answers

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 ...

**11**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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

**0**answers

39 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{...

**7**

votes

**0**answers

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 ...

**2**

votes

**1**answer

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 ...

**8**

votes

**1**answer

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 ...

**21**

votes

**4**answers

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 $\mathbb{K}$, $a$...

**2**

votes

**0**answers

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 $(a, x)= 0 = (x, b) \in H^2(F, \mathbb{F}_2)$ ...

**0**

votes

**1**answer

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 $y$ ...

**2**

votes

**3**answers

412 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 ...

**5**

votes

**0**answers

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 ...

**13**

votes

**3**answers

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 ...

**5**

votes

**0**answers

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 $p$ ...

**3**

votes

**0**answers

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 ...

**1**

vote

**0**answers

76 views

### Exceptional primes in Kummer-Dedekind theorem

Suppose that $A$ is a Dedekind domain with fraction field $K$, $L$ is a finite separable extension of $K$, and $B$ is the integral closure of $A$ in $L$. Suppose that $t$ is a primitive element for $L/...

**1**

vote

**0**answers

72 views

### “Algebrazing” canonical subgroups of elliptic curves

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...

**1**

vote

**2**answers

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 ...

**2**

votes

**1**answer

174 views

### Fundamental Units in Totally Real Cubic Fields

How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with ...

**2**

votes

**1**answer

152 views

### Character group of the multiplicative rationals

I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it here on math stack exchange first.
Let $\mathbb{Q}^{\times}$ be the multiplicative ...

**3**

votes

**1**answer

189 views

### Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.
Suppose that $p$ is ...

**10**

votes

**1**answer

237 views

### On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$

Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43}...

**2**

votes

**3**answers

352 views

### Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...

**9**

votes

**2**answers

553 views

### Number of polynomials whose Galois group is a subgroup of the alternating group

Let $f = x^n + a_{n-1}x^n + \cdots + a_0$ be a monic polynomial of degree $n \geq 2$ with integer coefficients. By $\text{Gal}(f)$ we mean the Galois group over $\mathbb{Q}$ of the Galois closure of $...

**1**

vote

**0**answers

79 views

### Why is the kernel of an algebraic Hecke character open in the ideles?

I've been reading about algebraic Hecke characters, and how one obtains one dimensional $p$-adic representations from them. I have a question about why the kernel of a map defined on the ideles is ...

**24**

votes

**0**answers

289 views

### Does every ring of integers sit inside a monogenic ring of integers?

Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has $\mathcal{...

**1**

vote

**0**answers

147 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**0**

votes

**0**answers

105 views

### More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings?
Can Number Field Sieve technique be applied here?

**4**

votes

**0**answers

63 views

### An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.
Is ...

**4**

votes

**2**answers

281 views

### The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/...

**7**

votes

**1**answer

212 views

### If two Hecke characters cut out the same field, are they Galois conjugates?

First question on MathOverflow, I hope it is appropriate for this site. There are two related questions.
Let $K$ be a number field, $G_K = Gal(\overline{K}/K)$, $p$ a prime, and
$$\chi_1,\chi_2:G_K\...

**5**

votes

**2**answers

202 views

### Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...

**3**

votes

**0**answers

118 views

### How to Taylor series expand at the prime at infinity

Given a rational number, one can find a Taylor series expansion with respect to any $p$-adic valuation, as covered in Gouvea's introductory text on $p$-adic numbers. My question is how does one do ...