# Questions tagged [rational-points]

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181
questions

7
votes

1
answer

330
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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

224
views

### 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{...

4
votes

1
answer

240
views

### 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

201
views

### 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}(...

7
votes

1
answer

381
views

### 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 ...

4
votes

1
answer

242
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

244
views

### 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

237
views

### 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 = \{...

2
votes

1
answer

236
views

### $2$-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, ...

4
votes

0
answers

135
views

### 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

141
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 ...

7
votes

2
answers

415
views

### 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 ...

1
vote

0
answers

65
views

### 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' $. ...

0
votes

1
answer

142
views

### 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 $ ...

2
votes

1
answer

78
views

### 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

222
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

82
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

306
views

### 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

186
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 $Q$?

5
votes

2
answers

494
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 ...

6
votes

1
answer

203
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 ...

2
votes

0
answers

227
views

### Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form
$$
S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\}
$$
where $...

4
votes

0
answers

176
views

### Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...

1
vote

0
answers

132
views

### Brauer-Manin obstruction and affine curves

I'm looking for references that can justify to what extent is the following statement true:
Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...

6
votes

0
answers

168
views

### Rational points on varieties whose anticanonical bundle is nef but not ample

Is the following plausible?
"If $X$ is a variety over $\mathbf{Q}$ whose anticanonical bundle $L$ is nef but not ample, there is a number field $K$ such that $X(K)$ contains an infinite set of ...

0
votes

1
answer

125
views

### Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$

Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$,
define
$E_a : x^3+a x z^2=y^2 z$
Let $B= \lfloor 2 \sqrt{p}\rfloor$
Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...

-1
votes

1
answer

202
views

### Bounds for the number of points on projective hyperelliptic curves over finite fields

Let $C$ be projective hyperelliptic curve over finite field $K$.
What are bounds for the number of points $\#C(K)$?
The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are
not smooth ...

5
votes

0
answers

216
views

### 2-descent on elliptic curves, and units modulo squares of units

Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...

10
votes

0
answers

194
views

### Are the nonnegative rationals diophantine with only two quantifiers?

Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...

4
votes

1
answer

368
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....

1
vote

0
answers

121
views

### Properties of pointless projective curves over finite fields?

Probably not research level, feel free to downvote.
We got construction of bounded degree projective curves
with no points over finite fields. This construction generalizes to higher dimension.
One of ...

4
votes

1
answer

355
views

### 3-, 6-, 12-descent for Z2xZ6 elliptic curves

We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...

6
votes

2
answers

267
views

### Can every set of points with rational distance squares be isometrically embedded in $\Bbb Q^d$?

Suppose we are given a finite family of points $p_1,...,p_n\in \Bbb R^d$, so that any two points have a rational distance square, that is,
$$\|p_i-p_j\|^2\in\Bbb Q,\quad\text{for all $i,j\in\{1,...,n\}...

1
vote

1
answer

200
views

### The smooth completion of a curve

Let $C$ be a smooth geometrically integral affine curve. This question concerns the smooth completion $C_1$ of $C$, both defined over a number field $k$.
We know that given any smooth projective ...

8
votes

4
answers

1k
views

### Status of $x^3+y^3+z^3=6xyz$

In
Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML
the author has studied the Diophantine equation
\begin{equation}
x^3+y^...

11
votes

2
answers

610
views

### 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 ...

2
votes

0
answers

162
views

### Finding rational points via birational map

Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$
and let $\overline{C}$ denote the projective closure of $C$. For ...

3
votes

1
answer

192
views

### Distribution of the rank of $y^2=x^4+x+b^2$

For positive integer $b$ define the curve $C_b : y^2=x^4+x+b^2$.
$C_b$ is genus one and has the rational points: $(0,\pm b),(-1,\pm b)$
and one more point from the reciprocal of the polynomial y=0
...

12
votes

0
answers

569
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 ...

2
votes

1
answer

154
views

### Is every sufficiently general monic quartic rational square infinitely often?

Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...

2
votes

2
answers

530
views

### A new simple formula is needed

The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$.
The SageMath/Python code below produces a list of small fractions $a$ for ...

5
votes

0
answers

172
views

### Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?

Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...

5
votes

1
answer

481
views

### rational points of a hyperelliptic curve of genus 3

Let $K=\mathbb{Q}(\sqrt{-1}).$ I have the following hyperelliptic curve of genus 3:
$$ C : y^2 = (x^2-x+1)(x^6+x^5-6x^4 -3x^3+14x^2-7x+1) $$
I want to find $C(K)$. My first attempt was to compute the ...

16
votes

0
answers

232
views

### Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?

In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...

2
votes

1
answer

161
views

### Criterion for existence of integral points on an elliptic curve

Is there a criterion for the (presumably infinite) set of $D \in \mathbb{Z}\setminus \{0\}$ such that
$$Dy^2 = x^3-1728$$
has an integral point over $\mathbb{Q}$ with $y \neq 0$? I'd also be ...

17
votes

1
answer

307
views

### Why should the number of $\mathbb{F}_q$ points on degree $d$ curves $C\subset \mathbb{P}_{\mathbb{F}_q}^n$ decrease as $n$ increases?

This question concerns some counterintuitive results (to me at least) regarding the number of points on a projective curve over a finite field. Namely, if one fixes the degree of the curve, but ...

8
votes

0
answers

124
views

### Distribution of rational points in the real locus of a planar algebraic curve

Let $C$ be a smooth projective geometrically connected curve over $\mathbb{Q}$. Assume that $g(C)=3$ and that $C$ is not hyperelliptic. Then the canonical sheaf defines a closed immersion $C\to\mathbb{...

3
votes

1
answer

317
views

### Finding $K$-rational points on $X_0(35)$

Let $K=\mathbb{Q}(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(35)$?
Recall that $X_0(35)$ is a hyperelliptic curve of genus 3 and has the simplified affine model:
\...

3
votes

2
answers

170
views

### primes of multiplicative reduction for elliptic curves with rational $\ell$-torsion

Very recently, I made an observation from scanning lists of elliptic curves on the LMFDB that leads me to the following (unproven) statement:
Fix $\ell \in \{5, 7\}$. Let $E$ be an elliptic curve ...

0
votes

1
answer

451
views

### How I can prove or disprove that $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1$ has solutions in rationals? [duplicate]

The motivation of this question is to look if there is such solution in rational number to the identity which mentioned here, I have done many attempts using Wolfram Alpha to find such pairs of ...