# Questions tagged [rational-points]

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195
questions

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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 $...

6
votes

1
answer

731
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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
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1
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170
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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
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1
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132
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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} \}.$...

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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 ...

4
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196
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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
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1
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45
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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 ...

2
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144
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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
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207
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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
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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 ...

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127
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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
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118
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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 ...

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### Rational points involved in the logistic, or sigmoid, function

$\text{“}$Everybody knows$\text{”}$ that $(0,\,1/2)$ is the only rational point on the graph of the sigmoid function $y = \dfrac 1 {1+e^{-x}}$.
$\text{“}$Somewhere on the internet$\text{”}$ someone ...

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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)\...

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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
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1
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242
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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{...

4
votes

1
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277
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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\...

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220
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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}(...

7
votes

1
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576
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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 ...

4
votes

1
answer

404
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### 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 ...

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281
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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

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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 = \{...

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1
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283
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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, ...

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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 ...

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1
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153
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### 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 ...

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votes

2
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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 ...

1
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0
answers

171
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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' $. ...

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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 $ ...

2
votes

1
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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 ...

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### 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. ...

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0
answers

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### 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
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1
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333
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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 ...

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### 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 $Q$?

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### 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 ...

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votes

1
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242
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### 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 ...

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0
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### 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 $...

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0
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### 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 ...

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### 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 ...

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### 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 ...

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1
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129
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### 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\...

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### 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
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0
answers

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### 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 $K_f := \mathbb{Q}_p[x]/(f(x))$, ...

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### 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 ...

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1
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### 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
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0
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129
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### 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

385
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### 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 $\...

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2
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283
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### 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\}...

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1
answer

260
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### 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 ...

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4
answers

2k
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### 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
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2
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645
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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 ...