The rational-points tag has no usage guidance.

**13**

votes

**1**answer

277 views

### Number of height-limited rational points on a circle

Consider origin-centered circles $C(r)$ of radius $r \le 1$.
I am seeking to learn how many rational points might lie on $C(r)$,
where each rational point coordinate has height $\le h$.
For example, ...

**5**

votes

**0**answers

88 views

### Counting square zero forms over finite fields

Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring.
Is ...

**5**

votes

**0**answers

153 views

### A relative version of Hensel's lemma?

Let $k$ be a $p$-adic field with integer ring $\mathcal{O}_k \subseteq k$, maximal ideal $m_k \subseteq \mathcal{O}_k$ and residue field $\mathbb{F}_q = \mathcal{O}_k/m_k$. Let $X$ be a smooth, ...

**2**

votes

**2**answers

154 views

### Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 ...

**2**

votes

**0**answers

148 views

### Rank of the Jacobian of twists of hyperelliptic curves

Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...

**5**

votes

**1**answer

251 views

### Transcendental distance sets

Define a set $S \subset \mathbb{R}^d$ as a
transcendental distance set if the distance between any pair of
distinct points of $S$ is transcendental.
For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots ...

**15**

votes

**1**answer

921 views

### Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states:
For every irrational number $\alpha$, there are infinitely ...

**4**

votes

**2**answers

881 views

### Find all rational solutions of this diophantine-equation?

Now, today, my friend tell me this problem was posted by American Mathematical Monthly (Vol. 111, No. 2 Feb., 2004), p. 165 by Wu wei Chao ,and It is said that this problem is unsolved, until now. ...

**8**

votes

**0**answers

143 views

### Distribution of Mordell–Weil ranks of higher genus curves

By "nice curve", I mean a smooth, projective, geometrically integral curve over $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Jac}{\operatorname{Jac}}\Q$ with at least one $\Q$-rational point. The ...

**6**

votes

**0**answers

104 views

### Endomorphism algebras of abelian surfaces with real multiplication

Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...

**5**

votes

**1**answer

166 views

### Average height of rational points on a curve

I am seeking a formalism to define the average height of
the rational points on a curve. This is straightforward
if the number of points is finite, but (to me) not straightforward
when the rational ...

**4**

votes

**2**answers

300 views

### Find all possible rational values of a parametric quartic such that it is reducible

Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 ...

**3**

votes

**2**answers

293 views

### cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:
1) is $X(\mathbb{Q})$ an empty set ?
2) is $X(\mathbb{Q})$ a finite (non empty) set ...

**13**

votes

**1**answer

588 views

### Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

**2**

votes

**0**answers

169 views

### The uniform boundedness of rational torsion for traceless abelian surfaces over a function field

The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the ...

**7**

votes

**1**answer

465 views

### rational points of a hyperelliptic curve

I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...

**10**

votes

**1**answer

242 views

### Schoenberg's Rational Polygon Problem

"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by ...

**5**

votes

**1**answer

225 views

### Maximum number of general-position points with mutual rational distances?

Richard Guy has shown that there are six points in the plane—no three collinear,
no four cocircular—such that all interpoint distances are rational.
Guy, Richard. Unsolved Problems in ...

**4**

votes

**1**answer

279 views

### What is the complexity of finding an integral point on an elliptic curve?

Let $E$ be an elliptic curve over rational numbers. We know that the set of integral points on $E$ is finite. What is the complexity of finding a point $P\in E(\mathbb{Z})$?
Indeed I'm trying to find ...

**6**

votes

**2**answers

402 views

### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...

**10**

votes

**3**answers

359 views

### Circles avoiding rational points of height $\le h$

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates ...

**1**

vote

**0**answers

159 views

### Coarse moduli spaces and rational points [closed]

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...

**10**

votes

**1**answer

250 views

### What is our current knowledge on the structure of J_0(N)(Q) and J_1(N)(Q)

The question in the title naturally breaks up in two parts, namely the torsion part and the rank part. I already read about some results on both the torsion and the rank part. And I want to know ...

**12**

votes

**2**answers

357 views

### Existence of local sections

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...

**6**

votes

**1**answer

439 views

### Is the following consequence of the Lang conjecture known?

This came up in a discussion with a colleague of mine, who studies PDEs. He was asking for a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$ such that, for all but finitely many $n$, the ...

**1**

vote

**0**answers

96 views

### Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...

**15**

votes

**1**answer

478 views

### Is the perimeter of an ellipse with integer axes irrational?

Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 ...

**3**

votes

**2**answers

140 views

### Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
...

**7**

votes

**0**answers

173 views

### Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...

**0**

votes

**0**answers

120 views

### Existence of a curve with no points over finite separable field extensions

Does there exist a field $K$, and a smooth projective geometrically connected curve $C$ over $K$ such that, for all finite separable field extensions $L/K$ the curve $C$ has no $L$-rational points?
I ...

**9**

votes

**2**answers

331 views

### Rational points on circular spirals

Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...

**1**

vote

**1**answer

174 views

### Infinite residue field extensions and algebraic closure of residue fields

Let $X$ be a $K$-scheme of finite type over a field $K$, let $L$ be an extension field of $K$, let $X_L := L \times_K X$, and let $p:X_L \rightarrow X$ be the projection. For each $x \in X_L$ we get ...

**2**

votes

**1**answer

212 views

### Closed points of field extension of k-scheme under projection

I really couldn't figure out the answer to the following question: Let $X$ be a scheme of finite type over a field $k$ and let $K$ be an extension field of $k$. Let $X_K := K \times_k X$ be the base ...

**2**

votes

**1**answer

360 views

### Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$

In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$.
In each case the x coordinates are ...

**27**

votes

**6**answers

1k 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.
...

**12**

votes

**1**answer

801 views

### Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...

**4**

votes

**1**answer

258 views

### Rational points on $X_0(15)$

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...

**0**

votes

**2**answers

266 views

### “rationality” of divisors

Let $X$ be a smooth projective variety over some field $k$. Then each closed point $x$ has an associated residue field $k(x)$ which is a finite extension of $k$ and a point is rational when $k(x)=k$.
...

**6**

votes

**1**answer

327 views

### Pick's Theorem for rational points of bounded height

I wonder if the various lattice-point theorems, such as
Pick's Theorem or
Minkowski's Lattice Theorem,
have been generalized to the collection of points
with rational coordinates no more than height ...

**15**

votes

**2**answers

804 views

### Identifying Ramanujan's integer solutions of x^3+y^3+z^3=1 among Elkies' rational solutions

In his Lost Notebook, Ramanujan exhibits infinitely many integer solutions to $x^3+y^3+z^3=1$. On his webpage (http://www.math.harvard.edu/~elkies/4cubes.html), Elkies determines all rational ...

**5**

votes

**2**answers

247 views

### Solving for special rational triangles

I ran into a need for isosceles triangles that (1) have the two equal
integer side lengths $a$ (but the base $x \in \mathbb{R}$),
and (2) the apex angle $\gamma$ is a rational multiple of $\pi$.
...

**10**

votes

**1**answer

322 views

### What evidence is there that $\mathbb{Q}^{ab}$ is ample?

A field $K$ is called ample if every smooth curve over $K$ that has a $K$-point has infinitely many $K$-points. Examples include fraction fields of henselian rings. (This is far from trivial, but was ...

**11**

votes

**2**answers

500 views

### Geometrically unirational varieties that are not unirational

By a variety over a field $k$, I mean a scheme that is separated and
of finite type over $k$. I indicate changes of the base ring by
subscripts.
Does there exist a smooth and projective variety $V$ ...

**11**

votes

**1**answer

299 views

### Can a harmonic number be a rational number for non-integer rational argument?

Define harmonic numbers for a complex argument $z$ as $H_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$.
For $n\in\mathbb{N}$, $H_n$ are usual harmonic numbers $\sum^n_{k=1} k^{-1}$ . They are ...

**15**

votes

**1**answer

358 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...

**1**

vote

**1**answer

219 views

### Tricks to produce examples of hypersurfaces with index greater than $1$

Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in ...

**0**

votes

**0**answers

157 views

### Deformation of rational points in a family

Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$-rational point. Fix a pair ...

**6**

votes

**1**answer

346 views

### Simple field extension and rational points

Let $F$ be an infinite field and $f$ a homogeneous form on $F$ such that $f$ has no non-trivial zero in $F$. Let $F'$ be a finite extension of $F$ such that $f$ has a non-trivial zero in $F'$. Is it ...

**10**

votes

**5**answers

2k views

### Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.
Q1.
Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e. does $S$ include a ...

**7**

votes

**1**answer

382 views

### Rational points on surfaces of general type

The weak Lang conjecture asserts that rational points on a variety of general type defined over $\mathbb{Q}$ are not Zariski dense (same replacing $\mathbb{Q}$ with a number field). This one is proved ...