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3
votes
0answers
120 views

Is Eisenstein's map from binary cubic forms really injective when $\Delta > 0$?

The four binary cubic forms, homogeneous cubics of the form $C_j(x,y) = a x^3 + b x^2 y + c x y^2+d y^3 = (a,b,c,d)$, $C_1 = (1,3,-15,-23)$, $C_2 = (1,3,-51,-203)$, $C_3 = (1,3,-33,-113)$, $C_4 = ...
13
votes
1answer
448 views

Growth of the “denominator” of powers of an algebraic number

Let $z \in \overline{\bf Q}$ be an algebraic number. Define the "denominator" of z to be the least natural number $n$ such that $nz$ is an algebraic integer. By a rather ad hoc argument (playing ...
6
votes
0answers
61 views

Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
4
votes
1answer
330 views

About principal ideal theorem in number fields

I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$. In this case, there is a well-known result that "every ambiguous class in the class group ...
3
votes
2answers
303 views

Cohen-Lenstra Heuristics reference

I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the ...
5
votes
1answer
283 views

Minimal polynomial of sums of roots of unity with constant term $\pm1$

Let prime $p$ and given $\zeta_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of $x = \zeta_p + \zeta_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of ...
1
vote
1answer
125 views

Algebraic numbers abhorrent to cyclotomic fields

Consider an algebraic number $\alpha$, which can be taken to be an integer. With $\deg\alpha$ a prime number, one can easily arrange that to be such that all powers $\alpha^n$ to be of the same ...
7
votes
1answer
302 views

When can number rings be spanned (as $\mathbb{Z}$-modules) by units?

Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Define $R \subset \mathcal{O}$ to be the set of all $\mathbb{Z}$-linear combinations of units. Since the product of two units ...
-1
votes
1answer
168 views

A characterization of the module function on a locally compact division ring

The same question was asked in Math StackExchange about 3 months ago. Since nobody has answered to it, I would like to post it here. References: Weil's Basic Number Theory(denoted by BNT). ...
9
votes
1answer
429 views

The Dissertation of F. J. van der Linden

Does anyone have access to the 1984 dissertation of Franciscus Jozef van der Linden under Hendrik Lenstra? It is called Euclidean Rings with two infinite primes. The theory is that this has the ...
3
votes
0answers
162 views

Euclidean real quadratic fields

It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic ...
1
vote
1answer
174 views

Number Field Sieve for factorization with non-monic non-linear polynomial. Can't understand calculating ideal valuations

There is a paper "Factoring integers with the number field sieve" (download it here, for example). I can't understand how they reason the correctness of computing ideal valuations in the case of ...
10
votes
1answer
370 views

primes represented by an indefinite binary quadratic form

Suppose I have a form $$ f(x,y) = a x^2 + b x y + c y^2, $$ with $a,b,c$ integers, $\gcd(a,b,c)=1$ and $\Delta = b^2 - 4 a c > 0,$ but $\Delta \neq n^2$ for any integer $n.$ Do there exist ...
5
votes
0answers
184 views

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
1answer
265 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 ...
5
votes
0answers
161 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 ...
2
votes
1answer
55 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 ...
2
votes
1answer
360 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
1answer
180 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 ...
2
votes
0answers
148 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 ...
1
vote
1answer
207 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   $C$.   ...
8
votes
2answers
361 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, ...
1
vote
2answers
185 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.
2
votes
1answer
179 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 ...
5
votes
0answers
133 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 ...
1
vote
1answer
514 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 ...
0
votes
0answers
85 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 ...
3
votes
0answers
209 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} ...
3
votes
0answers
113 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 - ...
1
vote
0answers
54 views

Isogenies in multidiensional 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 $F$. ...
1
vote
1answer
155 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 ...
3
votes
1answer
182 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 ...
2
votes
2answers
181 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 ...
27
votes
6answers
888 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. ...
3
votes
1answer
282 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 ...
6
votes
2answers
446 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 ...
0
votes
1answer
340 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 ...
6
votes
2answers
279 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: ...
1
vote
0answers
107 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), ...
7
votes
1answer
249 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? ...
8
votes
1answer
328 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 ...
3
votes
1answer
210 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 ...
0
votes
0answers
70 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 ...
3
votes
1answer
117 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 ...
5
votes
1answer
396 views

Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...
2
votes
2answers
159 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 ...
2
votes
0answers
104 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 ...
10
votes
1answer
790 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 ...
15
votes
1answer
439 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!
0
votes
1answer
205 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...