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### Generating congruence subgroups of SL_n over totally imaginary number rings

Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the ...

**2**

votes

**1**answer

290 views

### Some puzzles about the three conditions in a paper of D.Berend

Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend.
I am puzzled by the three necessary and sufficient conditions given there.
Could you provide me with some concrete ...

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votes

**0**answers

169 views

### When is a Number Ring generated by its Norm-1 elements?

In exercise 1.2.11. of Bump's Automorphic Forms and Representations book, he deals with real quadratic fields $K$ in which $\mathcal{O}_K$ is generated (as a ring) by its norm-1 units. In this case ...

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votes

**1**answer

60 views

### Action of GL(2,O_k) on 1d subspaces of (O_k)^2

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}_k$ (in other words, $M$ is a projective module such that $k ...

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votes

**1**answer

467 views

### Prime splitting in cubic field, congruence [closed]

Let $K$ be a cubic Galois extension of $\mathbb{Q}$.
I wonder if we can find a congruence for prime $p$ such that $p$ does not split completely in $K$. I know that we can do this for quadratic ...

**4**

votes

**1**answer

209 views

### Reduction mod $p$ of units in a ring of integers

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which ...

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152 views

### Lang's preprint “Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure”

I am trying to find the following preprint of Serge Lang, which supposedly discusses his C1 conjecture:
"Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure".
I have not ...

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vote

**1**answer

232 views

### Rational points of non-rational curves

An algebraic curve (in this question) is the zero set $C = f^{-1}(X\ Y)$ of any polynomial $f\in\mathbb R[X\ Y]$; we say then that $f$ represents $C$. ...

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votes

**2**answers

364 views

### property of trace modulo $n$

I recently noticed an interesting (at least to me) property of the trace but have been unable to prove it.
Let ${\mathbb K}$ is an algebraic number field with ${\mathcal O}$ as its ring of integers, ...

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vote

**2**answers

221 views

### Question about ring of integers of cyclotomic field [closed]

Let $\zeta=e^{\frac{2\pi i}{p^n}}$, $p$ is an odd prime. Is $\mathbb{Z}[\zeta]$ a UFD?
Thank you for watching.

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votes

**1**answer

185 views

### RefReq: Algorithms for standard operations in Algebraic Number theory

Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code ...

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162 views

### Salem and Perron polynomials

If $P(t)\in \mathbb{Z}[t]$ is a polynomial, let $d$ be its degree and let $P_{*}(t)$ denote its reciprocal polynomial, i.e. $P_{*}(t) := t^d\, P(1/t)$.
Let $Q_n(t) \in \mathbb{Z}[t]$ be a polynomial ...

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vote

**1**answer

558 views

### An integral representation of the Riemann zeta function

I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper.
I am unable to recognize where this comes from or what is the general expression for values other than ...

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93 views

### Derivative of a function related to Dedekind zeta function

Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$.
From the ...

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244 views

### Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
...

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**0**answers

123 views

### Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to:
If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...

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vote

**0**answers

108 views

### Isogenies in multidimensional formal groups

Let $K/\mathbb{Q}_p$ be a local field, $A$ the ring of integers of K, $\pi$ a uniformizer element for $A$, $F$ an n-dimensional formal group with coefficients in $A$ and $f$ an endomorphism of $F$. ...

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vote

**1**answer

176 views

### Deciding if the largest absolute value real root lies in a cyclotomic extension

Given an algebraic equation of degree $n$ of form: $$x^{n} - a_{n-1}x^{n-1} - a_{n-2}x^{n-2} - \dots - a_{0} = 0$$ where each $a_{i} \in \Bbb Q^{+}$ and atleast one positive root, how does one decide ...

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**1**answer

207 views

### Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...

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**2**answers

195 views

### What is the exact meaning of the real period in the $p$-adic formulation of BSD?

Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism ...

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957 views

### Patterns among integer-distance points

Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...

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**1**answer

383 views

### Prime ideals in the ring of algebraic integers

Let $m(x) = x^n + a_{n-1}x^{n-1} + \dots + a_1 x+ a_0$, $a_i \in \mathbb{Z}$, be an irreducible polynomial over $\mathbb{Q}$ and $K = \mathbb{Q}(x) / {m(x)\mathbb{Q}(x)}$. K is an algebraic number ...

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469 views

### Is the infimum of Salem numbers > 1?

BACKGROUND
A Salem number is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their ...

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**1**answer

342 views

### How to do such a partitioning?

Assume:
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N}
$$
and,
$$
f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j w^{(p_i-p_j)l}
$$
I am going to ...

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330 views

### Class numbers of orders

Consider an order $R$ in a number field $L$. Let $C_R$ be the set of $R$-fractional ideals modulo $L^\times$. Let $O$ be the maximal order in $L$, and $C_O$ be the class group of $O$.
My question: ...

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**0**answers

138 views

### Reference request for a basic result on relative differents & discriminants

I am looking for a better reference for the results in this extremely short and elementary paper:
Tôyama, Hiraku,
`A note on the different of the composed field',
Kōdai Math. Sem. Rep. 7 (1955), ...

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**2**answers

391 views

### Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?
Are there infinitely many? Finitely many? ...

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**1**answer

417 views

### Fundamental units of imaginary quartic fields

Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...

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**1**answer

236 views

### $\ell$-conductor of a two-dimensional $\ell$-adic Galois representation

Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...

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77 views

### Divisor bounds of ideals in number fields

Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers).
Denote by $d(I)$ the number of ideals that divide $I$.
So if $I= \prod_{i=1}^k p_i^{e_i}$ is the ...

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votes

**1**answer

124 views

### How to estimate a local hilbert samuel funcion

Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...

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**1**answer

461 views

### Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...

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**2**answers

194 views

### On Cubic Non-Residues Modulo a Prime [closed]

What is a good test for identifying cubic non-residues/residues and higher power non-residues/residues modulo a prime $R$ in terms of computational complexity?
Given $M$ and $N$, is there a good way ...

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116 views

### What is the real subring of a ring of cyclotomic integers?

I am looking at tilings whose vertices lie in a ring of cyclotomic integers. These tilings are of interest as they can have interesting scaling properties or be substitution tilings. Interesting ...

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**1**answer

1k views

### Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?

From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...

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**1**answer

452 views

### Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?

By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!

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**1**answer

225 views

### For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...

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vote

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158 views

### Lower Degree Elements in an Algebraic Number Field

Fix an algebraic integer $\alpha$ of degree $n$
such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields.
(We can assume $K$ is Galois with non-simple Galois group.)
This ...

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### classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...

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623 views

### Quintic polynomial solution by Jacobi Theta function.

Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...

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78 views

### Is the equidissection spectrum closed under addition?

If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area?
(I'm editing after realizing that my conjecture that a ...

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**1**answer

316 views

### Extensions of Galois representations

Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ...

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### If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous and amazingly tricky proof says that if we ...

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**1**answer

720 views

### Questions about the proof of Stickelberger's theorem on discriminants

I was going through the proof of Stickelberger's theorem about discriminants in the book 'Algebraic Number Theory' by Richard A. Mollin, and I am having some problems in understanding the proof. I ...

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417 views

### Simplifying an algebraic integer expression

I have an expression where the variables are algebraic integers:
$p4 = \frac{p12 - p41 \cdot p21}{p22}$
p12 is degree 48 and p22 is most likely degree 48 too. p41 is degree 32 and p21 is degree 24. I ...

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946 views

### Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$.
Question: Let $p$ be an ...

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88 views

### Decompositions of representations of pro-p groups

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...

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137 views

### ramification of discrete valuation field

Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb ...

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84 views

### points in $V(\bar K \otimes_{\bar Q} \bar L)$ rational over tensor product of fields

Let V be a variety over a number field, and let K and L be two algebraically closed
What is known about the points of $V(\bar K \otimes_{\bar Q} \bar L )$ ?
Are there results claiming that points in ...

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739 views

### how to visualize the class number of an imaginary quadratic field?

Let me detail the title of the question. I'm trying to give students an intuition of what the class number is.
Let $K=\mathbb{Q}(\sqrt{-d})$, with $d>0$ a square-free integer, be a quadratic ...