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-2
votes
0answers
37 views

Two rational and one irrational root of a cubic? [on hold]

Let $p(x)=a_3x^3+a_2x^2+a_1x+a_0$, with $a_i\in\mathbb{Q}$. Is it true that if two of the roots of $p(x)$ are in $\mathbb{Q}$, then the third is as well?
2
votes
0answers
141 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 ...
-2
votes
0answers
120 views

Bounded pythagorean triples [migrated]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
5
votes
1answer
245 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
1answer
882 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
2answers
855 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
0answers
134 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
0answers
99 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
1answer
156 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
2answers
293 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
2answers
290 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 ...
12
votes
1answer
580 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
0answers
168 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
1answer
426 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
1answer
236 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
1answer
224 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
1answer
277 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
2answers
395 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
3answers
352 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
0answers
158 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
1answer
246 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
2answers
354 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
1answer
438 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
0answers
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
1answer
477 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
2answers
138 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
0answers
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
0answers
118 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
2answers
330 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
1answer
171 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
1answer
211 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
1answer
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
6answers
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
1answer
785 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
1answer
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
2answers
261 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
1answer
318 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
2answers
797 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
2answers
245 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
1answer
321 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
2answers
498 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
1answer
296 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
1answer
355 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
1answer
217 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
0answers
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
1answer
345 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
5answers
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
1answer
376 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 ...
3
votes
0answers
287 views

Rational points and Tesla cards

I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8. Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves. ...
7
votes
1answer
298 views

Varieties with infinitely many etale covers and rational points

Let $X$ be a (smooth projective geometrically connected) variety over a field $k$. Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$. ...