Questions tagged [rational-points]
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202
questions
23
votes
3
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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.
...
2
votes
0
answers
110
views
Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
20
votes
3
answers
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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 ...
6
votes
0
answers
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Ranks of elliptic curves over cubic fields
We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations.
D. Jeon,...
19
votes
2
answers
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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 ...
2
votes
1
answer
272
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An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)
$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\Br(X)/{\Br(k)}$ being an infinite ...
4
votes
2
answers
297
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Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
8
votes
0
answers
224
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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 \...
4
votes
2
answers
295
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Algorithm for computing rational points if the rank of Jacobian is 0
Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? ...
3
votes
0
answers
157
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Smoothness of height in Manin conjecture
Set up: Let $K$ be a number field. Let $M_K$ be the places of $K$, and define the standard height on $\mathbb{P}^n(K)$ as
$$H([x_0, \cdots, x_n]) = \prod_{v \in M_K} \max\{|x_0|_v, \cdots, |x_n|_v\}$$
...
3
votes
1
answer
258
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Leading constant in Batyrev-Tschinkel's refinement of Manin conjecture
Background: Let $X$ be a Fano variety over number field $K$, where its anticanonical bundle $K_X^{-1}$ is ample. Let $i: X \to \mathbb{P}^n$ be the anticanonical embedding, where $K_X^{-m} \cong i_*O(...
0
votes
0
answers
41
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Educated guess for algebraic approximation
I found a very neat ancient hindi formula for approximating square roots using rational numbers. After doing some algebra on the formula, i came across with this recursive relation:
Given any number $...
4
votes
1
answer
839
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Does this conic have a rational point?
Consider the conic
$$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$
over the function field $\mathbb{Q}(u,v)$.
Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
5
votes
1
answer
191
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Fields of definition of conjugates
Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that ...
2
votes
1
answer
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A variant on the Fujita invariant
Let $X$ be a Fano variety over $\mathbb{C}$. Let $D$ be a divisor on $X$. Recall that the Fujita invariant of $D$ is defined to be
$$a(D) = \inf \{ t \in \mathbb{R} : K_X + tD \text{ is effective} \}.$...
2
votes
0
answers
190
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How dense is the set of rational points of a variety?
General question: Let $W$ be a proper subvariety of an irreducible affine variety $V/K$. Under what conditions do we know that $W(K)$ is a proper subset of $V(K)$?
If $K$ is finite, then one can bound ...
171
votes
3
answers
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Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
2
votes
1
answer
224
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Classification of quartic surfaces
Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\...
1
vote
1
answer
70
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Fixed points of rational continuous piecewise affine maps
Say that a compact convex polytope is rational if is the intersection of half-spaces whose bounding hyperplanes are the zero-sets of affine functions of the coordinates with rational coefficients. Say ...
1
vote
1
answer
148
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Geometry of contracted divisors
Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:...
7
votes
1
answer
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Reference request. Finiteness of the Selmer group
Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
5
votes
1
answer
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How to make Burnside's formula compatible with point counting for varieties over finite fields?
If $G$ is a finite group acting on a finite set $X$, we have Burnside's formula that counts the number of orbits $|X/G|$ as:
$$ |X/G| = \frac1{|G|} \sum_{g\in G} |X^g|,
$$
with $X^g$ being the set of ...
-1
votes
1
answer
128
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Does this quadratic system admit an integral or a rational solution?
Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
4
votes
0
answers
125
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Statistics about existence of rational points on a curve over $\mathbb{F}_q$
I wish to ask the naive question: if we write down a random curve $C$ over $\mathbb{F}_q$, what can be said about the probability that $C(\mathbb{F}_q)=\emptyset$?
Of course, this depends on the ...
7
votes
2
answers
568
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ℤ/18ℤ elliptic curves over cubic fields
I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of
D. Jeon, C. H. Kim, Y. Lee, Families of elliptic curves over cubic ...
11
votes
1
answer
639
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Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?
Consider the quartic system in four variables $a,b,c,d\in\mathbb R$:
$$-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac).$$
Does this system admit rational solution with $$abcd(c^2-d^2)(a^2-b^2)(a^2-c^2)(b^2-d^2)\...
7
votes
1
answer
490
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Field extensions over which algebraic varieties cannot acquire points
The following fact (slightly reworded here) is proven in this answer:
If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite type over $...
2
votes
1
answer
252
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Rational points on a special class of surfaces
Consider a smooth surface of the following form
$$
S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3
$$
over $\mathbb{Q}$, and set
$$
U_S = \{t' \in \mathbb{...
3
votes
1
answer
348
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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\...
4
votes
1
answer
237
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Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$
Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve
$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$
More precisely, $C$ is a twist of the modular curve $X_{0}(...
2
votes
1
answer
307
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$2$-isogenous to a curve in the Tate normal form
It is well-known that an elliptic curve $E$ that has a point of order $2$ and is represented as $E=[0,a,0,b,0]$ has a $2$-isogenous curve $E^\prime=[0,-2a,0,a^2-4b,0]$, see e.g. p. 507 in
A. Dujella, ...
6
votes
1
answer
690
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Singular curves of genus 1
Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$.
Is $C$ rational over $k$?
If $C$ is a plane cubic the answer is positive since we can ...
3
votes
1
answer
557
views
Number of points of a quadric hypersurface over a finite field
Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$.
By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...
3
votes
1
answer
296
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Smooth surfaces in positive characteristic
Let $K = \mathbb{F}_p$ be a field of positive characteristic $p > 0$. Consider a surface in $\mathbb{A}^3_K$ of the following form
$$
S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+...
4
votes
1
answer
290
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Del Pezzo surfaces of degree four and complete intersections of two quadrics
Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.
Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
4
votes
0
answers
173
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Rational points on ramified coverings of abelian varieties
Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is:
Suppose that $f(X(K)) \neq A(K)$, can ...
1
vote
1
answer
162
views
Space of rational conics
Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$.
Conisider the ...
0
votes
1
answer
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Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
1
vote
0
answers
241
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Surjectivity of a norm map over $ \mathbb{Q} $
Suppose $ (L/L') $ is an galois extension , where both fields are extension of $Q$ of $\dim n $ and $\dim n^{'}$ respectively.Suppose we consider the norm map $ Nr_{(L/L')} :L \rightarrow L' $. ...
2
votes
1
answer
149
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Linear subspace in quadric hypersurfaces over a field
Let $K$ be a field of characteristic different from two, and $Q\subset\mathbb{P}^{n+1}_K$ an $n$-dimensional smooth quadric hypersurface over $K$.
Suppose also that $Q$ has a $K$-point and so $Q$ is ...
3
votes
1
answer
251
views
Rationalizing and minimizing elliptic curve coefficients
I am working on elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ over quadratic fields. The curves are constructed using the model $E_1=[0,a,0,b,0]$ following the formulas on p. 13 of
L. ...
0
votes
0
answers
86
views
Computational tool for checking the existence of non-trivial rational zero of a cubic form
Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ...
0
votes
1
answer
350
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Systems of equations for elliptic curves without $3$-torsion
In his YouTube video New rank records for elliptic curves having rational torsion, Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of rank 14 and rank ...
5
votes
1
answer
497
views
Lines on quadric surfaces
Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
6
votes
2
answers
381
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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 $\pi$.
&...
4
votes
1
answer
406
views
Z2xZ6 elliptic curves with missing generators
By implementing the techniques described in and similar to
A. Dujella, J. C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z, arXiv, Number Theory [math.NT] (2013), arXiv:1306.0027v1
A....
12
votes
0
answers
656
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
11
votes
2
answers
666
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Z/8Z elliptic curve with a missing generator
We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in
A. J. MacLeod, A Simple Method for ...
5
votes
2
answers
548
views
Birational geometry over finite fields
I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
7
votes
1
answer
302
views
Geometrically rational variety over a finite field
Let $k=\mathbb{F}_q$ be a finite field, and let $X$ be a smooth projective variety over $k$. Suppose that $X_{\overline{k}}$ is birational to $\mathbb{P}^n_{\overline{k}}$, do we know
(1)If $X$ is ...