# Questions tagged [diophantine-geometry]

The diophantine-geometry tag has no usage guidance.

56
questions

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### Equidistribution of Hecke points and Steinitz classes

Let $K$ be a number field and $\mathcal{P}$ be a prime ideal of $K$. Let $k_{\mathcal{P}}$ be the residue field $\mathcal{O}_K/\mathcal{P}$.
Consider the following construction used very often in ...

2
votes

0
answers

111
views

### The connection of Faltings height and Tate module

Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...

1
vote

3
answers

188
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### A rational distance problem with (possibly) multiple solutions

Background: It is not known if any point exists on the XY plane that is at rational distance from all 4 vertices of a unit square lying on the XY plane (https://mathworld.wolfram.com/...

0
votes

0
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57
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### To tile the plane with mutually non-congruent rational triangles of equal area

We add a little to Tiling the plane with pairwise non-congruent rational triangles
Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "...

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0
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103
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### Which polygons allow partition into rational triangles?

A triangle with all side lengths rational is said to be a rational triangle. It is known - for example, Cutting the unit square into pieces with rational length sides - that the unit square allows ...

0
votes

1
answer

146
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### Almost Pell type equation

Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?

8
votes

0
answers

223
views

### Lattice point counts on the determinantal variety

I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$.
$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...

1
vote

2
answers

123
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### Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$

Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...

5
votes

1
answer

110
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### How should multiplicative height on projective space interact with automorphisms?

Background on heights
Consider $P = [a: b] \in \mathbb{P}^1(\mathbb{Q})$, where $a,b$ are coprime integers. We define the naive (multiplicative) height as
$$H(P) = \max \{|a|, |b|\}$$
We can change ...

2
votes

1
answer

231
views

### System of two linear Diophantine equations

Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system
$$
\left\lbrace\begin{array}{l}
\sum_{i=1}^nx_i = 3n; \\
\sum_{i=1}^n (2i-1)x_i = ...

7
votes

2
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707
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### Well known applications of Roth's theorem

Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places.
It is an ...

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0
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121
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### Height on $\mathbb G^n_m$ and Néron–Tate heights

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and let $L$ be a line bundle on $A$. Then a Néron–Tate height $\hat h_L$ can be defined by taking a model $\mathcal A$ of $A$ (over the ring ...

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112
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### Integral points in smooth cubic curves

Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...

1
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0
answers

76
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### Dyson's lemma implies index is small (in proving Roth's theorem)

I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...

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votes

1
answer

128
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### Does this quadratic system admit an integral or a rational solution?

Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...

3
votes

1
answer

338
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### Rational points of bounded height on a variety

I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...

6
votes

2
answers

683
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### Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$

Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...

3
votes

0
answers

103
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### Local expression of a quasi projective variety under finite morphism

I am studying the article Around the Chevalley-Weil Theorem by Zannier, Turchet and Corvaja and I am stuck in the following point: let $V$ and $W$ be two complex quasi-projective varieties and $\pi\...

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0
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137
views

### A specific Diophantine equation related to the congruent number question

Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...

3
votes

0
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425
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### Closed immersion hitting all the $\mathbb{Q}$-points

Let $i:X\to Y$ be a closed immersion of smooth projective varieties over $\mathbb{Q}$.
Assume that $Y(\mathbb{Q})$ is infinite and $X(\mathbb{Q})\to Y(\mathbb{Q})$ is surjective. Also assume that $X$ ...

0
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0
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77
views

### On four non-cocyclic integral points on ellipse

Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...

2
votes

0
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216
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### Have the following summations been studied before?

Suppose $q^2-4pr<0$, and consider the set of integral points
$$\mathcal Z=\{(X,Y)\in\mathbb Z^2:$$
$$px^2+qxy+ry^2+sx+ty+u=0\}$$
which lie on an ellipse. Then define $$M=\sum_{(X,Y)\in\mathcal Z}\...

6
votes

1
answer

225
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### How can the number of rational points depend on the choice of height function?

Let $V/\mathbb{Q}$ be a subvariety of $\mathbb{P}^n$. There are many plausible choices of height function, some differing only by constant factors: $\max |x_i|$ (for $(x_0,x_1,\dotsc,x_n)$, $\gcd(x_1,\...

0
votes

0
answers

280
views

### Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$

I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$.
Lattice point means integer coordinates and equation with integer means diophantine ...

1
vote

1
answer

198
views

### Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application

Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil<
r_1,r_2<\lceil1+\sqrt{p}\rceil$ and
$$r_1\equiv mac\bmod p$$
$$r_2\equiv mbd\bmod p$$...

15
votes

1
answer

439
views

### Looking for a paper on transfinite diameter by David Cantor

I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while:
Cantor D.: On an extension of the definition of ...

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0
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143
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### Looking for a paper by Vinberg

I have been reading about the Schinzel-Zassenhauss conjecture, and have been looking for the following reference:
Vinberg E.: On some number-theoretic conjectures of V. Arnold, Japanese J. Math., vol ...

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0
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132
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### Clarification regarding the definition of absolute height of an algebraic number

According to page 1-2 of this paper (https://arxiv.org/abs/0906.4286), Mahler has established the inequality
$$|\alpha_1 - \alpha_2| \geq H(\alpha_1)^{-(d-1)} \tag 1 $$
to be valid for all pairs of ...

1
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0
answers

55
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### Diophantine bound for homogeneous system under norm conditions for solutions and system

If $b_{ij}\in[0,2^t-1]\cap\mathbb Z$ holds at every $i\in\{1,\dots,k'\}$ and $j\in\{1,\dots,k\}$ where $1\leq k'\leq k$ holds then there are $x_1,\dots,x_k\in\mathbb Z:\sum_{j=1}^kx_jb_{1i}=0\wedge\...

0
votes

0
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132
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### Chinese remaindering to solve solvable diophantine equations

Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...

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1
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440
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### Cutting the unit square into pieces with rational length sides

The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational.
To cut a unit square into n (a finite ...

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### References for heights of algebraic or projective variety

In my research I am using the notion of a $\textbf{rational function f on a domain}\: U\subset \mathbb C^{n}$. By this I mean that the graph of $f$ is an analytic component of an affine variety $X$ ...

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### The Ax-Kochen isomorphism theorem and the continuum hypothesis

Let's recall that:
(1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \...

6
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0
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### *Why* is Bombieri-Pila uniform?

I am about to give a couple of lectures on Bombieri-Pila/the determinant method. Bombieri-Pila gives a bound $$|C(\mathbb{Q}) \cap B \cap \mathbb{Z}^2| \ll_{d,\epsilon} N^{1/d+\epsilon}$$ for $C$ a ...

1
vote

0
answers

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### Generalization of a result of Frey

In Proposition (2) in the paper [1], in below, it is proved that:
Assume that $K$ is a number field and that $C$ has infinitely many points of degree $\leq d$ over $K$. (Here $d\geq 2$ is an integer)...

1
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0
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### Maximum number of bounded primitive integer points in a zero-dimensional system

Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...

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0
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### Smallest integer lattice point by box measure in a polytope?

Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...

5
votes

0
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### Isomorphism classes of lattices

Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define
$$
V = \{x \in \mathbb R^6 \mid A \cdot x = 0\}
$$
and
$$
\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...

4
votes

0
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145
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### Number of nontrivial integral solutions to $f(x)=f(y)$

Let $f(x)\in\mathbb Z[x]$ be a nonconstant polynomial, and let $$g(x,y)=\frac{f(x)-f(y)}{x-y}\in\mathbb Z[x,y].$$ Let $N(B)$ denote the number of pairs of integers $(x_0,y_0)$ such that $1\le x_0,y_0\...

2
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0
answers

142
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### Genus Zero Diophantine Equations and Infinite Valuations

I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...

-1
votes

2
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229
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### On distribution of size of integer points in a subspace associated to a linear diophantine equation

Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$
n/4<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space of solutions to $ACa+ADb+BCc+BDd=0$ spanned by $3\times 4$ matrix
$$N=...

2
votes

0
answers

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### A question on a certain family of complete intersection varieties

Let $k$ be a field of characteristic zero.
Given integers $2 \leq s \leq r < n$, define the variety $X_n$ in $P_K^n$ with coordinates $y_0, \cdots, y_n$ and $K=k(u_0, \cdots, u_n)$ ( where $u_i$'...

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### Any simple concrete proof of Faltings theorem?

Are there simple proofs of some concrete special cases of Faltings's theorem? Any help would be appreciated.

13
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1
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514
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### The number of representations of an integer as the inner product of integral lattice points

I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an ...

2
votes

1
answer

439
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### Counting algebraic points of bounded height

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set
$$S(X;D,B)=\{\xi\in X(\...

5
votes

0
answers

602
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### Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?

This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma:
Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say
$a_{11}x_1+\dots+...

3
votes

0
answers

125
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### Finiteness of rational points on certain open subset of surfaces of general type

Let $X$ be a smooth surface of general type defined over a number field $k$.
The Bombieri-Lang conjecture asserts that there is a proper Zariski-closed set $Z$ in $X$ such that for any finite ...

2
votes

0
answers

656
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### Self intersection of theta divisor

I hope my question is not too basic here.
I have a very specific setting and I am trying to not use "big theorems" to prove a well known fact algebraically. I hope someone can give me a hint.
Let $J/...

39
votes

1
answer

2k
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### Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$

For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...

7
votes

2
answers

461
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### Obstruction to rationality of del Pezzo surfaces of degree 4

Let $X$ be a del Pezzo surface over a number field $k$. (A del Pezzo surface over $k$ is a smooth, projective, geometrically connected surface whose anti-canonical class $K_X$ is ample.) Let $d := K_X^...