Questions tagged [rational-points]
The rational-points tag has no usage guidance.
216 questions
8
votes
1
answer
535
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 ...
- 3,625
9
votes
1
answer
962
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 ...
- 151k
4
votes
2
answers
411
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 +...
- 441
6
votes
0
answers
438
views
Brauer-Manin obstruction to surfaces of Kodaira dimension 1
Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...
- 998
3
votes
2
answers
520
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 ...
- 205
14
votes
1
answer
1k
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.
- 345
3
votes
0
answers
282
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 ...
- 13.8k
12
votes
1
answer
2k
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. ...
- 441
11
votes
1
answer
702
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 a ...
- 151k
5
votes
1
answer
393
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 ...
- 151k
4
votes
1
answer
560
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 ...
- 369
9
votes
2
answers
792
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 ...
- 2,224
10
votes
3
answers
683
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 are ...
- 151k
1
vote
0
answers
211
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, ...
- 2,323
12
votes
1
answer
617
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 ...
- 2,224
12
votes
2
answers
424
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 ...
- 121
6
votes
1
answer
518
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 ...
- 4,745
2
votes
0
answers
120
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 ...
17
votes
1
answer
822
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 ...
- 151k
3
votes
2
answers
185
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 $C(r)$:
&...
- 151k
7
votes
0
answers
205
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 ...
- 151k
1
vote
0
answers
193
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 ...
- 51
9
votes
2
answers
449
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 ...
- 151k
1
vote
1
answer
543
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 ...
- 313
4
votes
1
answer
874
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 ...
- 313
2
votes
1
answer
387
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 ...
- 791
28
votes
6
answers
2k
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.
...
- 151k
24
votes
3
answers
3k
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.
...
- 151k
4
votes
1
answer
741
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 ...
- 295
0
votes
2
answers
400
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
667
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 ...
- 151k
19
votes
2
answers
1k
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 ...
- 6,397
6
votes
2
answers
392
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 $\pi$.
&...
- 151k
12
votes
1
answer
361
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 ...
- 5,929
11
votes
2
answers
791
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 $V$ ...
- 4,745
13
votes
1
answer
561
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 ...
- 1,699
25
votes
3
answers
994
views
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 faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
- 513
1
vote
1
answer
241
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 $\mathbb{P}^n_K$...
- 2,032
0
votes
0
answers
214
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 $(p,\...
- 2,032
6
votes
1
answer
489
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 ...
- 2,032
22
votes
5
answers
7k
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.
Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
- 151k
13
votes
1
answer
1k
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 ...
- 7,722
3
votes
0
answers
309
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.
...
- 791
9
votes
1
answer
549
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 $k$.
...
- 125
3
votes
1
answer
2k
views
Rational subspaces
In $\mathbb{R}^n$, we say that a linear subspace is rational if it admits a basis in $\mathbb{Q}^n$ (or equivalently in $\mathbb{Z}^n$). This means that $E\cap \mathbb{Z}^n$ is a submodule of $\mathbb{...
- 830
5
votes
2
answers
852
views
12 descent scripts for pari/gp
I'm looking around for scripts to facilitate 12 descent on Mordell curves, preferably in Pari/gp.
I understand that Magma implements this feature, but unfortunately this software isn't available to ...
- 791
15
votes
3
answers
3k
views
Rational Points on $y^2=x^3-86069^5$
The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an ...
- 791
20
votes
3
answers
2k
views
what is the maximum number of rational points of a curve of genus 2 over the rationals
Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...
- 209
1
vote
1
answer
133
views
are p-limits scales dense in the infinite musical scale of all rational frequencies?
In the wiki section on prime limit tuning, one reads:
...
- 7,853
13
votes
2
answers
572
views
Existence of points on varieties which avoid a given number field.
Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that
$L \cap K' = K$, and
$C(L) \neq \...
- 10.5k