Questions tagged [rational-points]

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5
votes
0answers
85 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 ...
2
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0answers
142 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....
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0answers
100 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
1answer
242 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 $\...
7
votes
2answers
250 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
1answer
146 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
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4answers
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^...
9
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2answers
407 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
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0answers
147 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
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1answer
166 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 ...
9
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0answers
360 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
1answer
139 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
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2answers
507 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
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0answers
169 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
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1answer
366 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 ...
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0answers
221 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
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1answer
152 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 ...
16
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1answer
283 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
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0answers
116 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
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1answer
263 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
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2answers
114 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
1answer
412 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 ...
2
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1answer
222 views

Perfect square quadratic expression

For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square. I start with $y^2=(5cx+100)(5cx-64c+36)$ and ...
4
votes
1answer
327 views

A generator needed for a Z/6 elliptic curve

We are searching for rank $8$ elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in A. Dujella, J.C. Peral, P. Tadić, ...
8
votes
1answer
764 views

Hard: One more generator needed for a Z/6 elliptic curve

We are searching for rank 8 elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in A. Dujella, J.C. Peral, P. Tadić, Elliptic ...
7
votes
2answers
354 views

Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

I am interested to know if a similar theorem that shows this answer of the post Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
2
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1answer
298 views

Density of rational points over finite fields, an estimate of Lang-Weil constant

Let $X\hookrightarrow\mathbb P^n_{\mathbb F_q}$ be a geometrically integral hypersurface over the finite field $\mathbb F_q$ of degree $\delta$. In order to estimate the number of its rational points, ...
2
votes
1answer
214 views

Resolved: Two more generators needed for a Z/6 elliptic curve

We are searching for the rank 8 elliptic curves with the torsion subgroup Z/6 using newly discovered families similar to Kihara's (Kihara's family is described in https://arxiv.org/pdf/1503.03667.pdf)....
7
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1answer
172 views

Subfields of Hilbertian fields

This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here : http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf My ...
3
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0answers
222 views

Integral points on affine varieties

Consider Siegel's theorem. It says that for a smooth affine algebraic curve $C$ over $\mathbb{Q}$ such that $g(C)>0$ any model $\mathcal{C}$ of $C$ over $\mathbb{Z}$ has finitely many $\mathbb{Z}$-...
7
votes
1answer
335 views

One more generator needed for a Z/6 elliptic curve

I am trying to find the next rank 8 curve with the torsion subgroup Z/6 using Kihara's family as described in https://arxiv.org/pdf/1503.03667.pdf. Meanwhile, I came across a curve generated by $t=629/...
3
votes
1answer
113 views

Connected sum of algebraic curves, handlebody decomposition, and induction on genus

Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...
4
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2answers
568 views

Pointless, non-singular, absolutely irreducible affine plane curves over finite fields

We think the following is true: For all sufficiently large primes $p$ and all natural $g \ge 1$, there exists affine plane curve $f(x,y)=0$ over $\mathbb{F}_p$ which is non-singular, absolutely ...
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0answers
161 views

Computing the genus of a plane curve

Let $b(x)=x^4 + 3x^3 + 3x^2 + 2x + 1$, and let $a(x)\in \mathbb Z[x]$ be a separable polynomial. Let $C$ be the plane curve defined by $(y^2+(x+x^2+x^3)a(x))^2-a(x)^2b(x)=0$. I would need to show that ...
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1answer
399 views

Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$? Do there exist ...
5
votes
4answers
296 views

Lattice points in a square pairwise-separated by integer distances

Let $S_n$ be an $n \times n$ square of lattice points in $\mathbb{Z}^2$. Q1. What is the largest subset $A(n)$ of lattice points in $S_n$ that have the property that every pair of points in $A(n)$...
4
votes
1answer
296 views

Submersion implies many rational points in image?

Let $A \colon V \to W$ be a surjective linear map (defined over $\mathbb{Z}$), inducing a projection $\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$. Let $X \subseteq \mathbb{P}(V)$ and $Y \...
6
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1answer
298 views

Breaking a morphism with generic fiber $\mathbb{F}_n$

Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch ...
3
votes
1answer
152 views

Solving system of multivariable algebraic equations over $\mathbb Q$ by reducing over $\mathbb F_p$

I try to solve the finite system of multivariable algebraic equations with coefficients from $\mathbb Q$. It would be sufficient for me to prove that there is only finite number of solutions over $\...
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0answers
87 views

Rational points of torsors over a separable closure

I already asked this question on Math Stack few days ago ( torsors over a separable closure ), but did not receive any answer, so I post it here. Let $G$ be a smooth linear algebraic group defined ...
2
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0answers
274 views

Rational point on variety over function field

This is one of the theorems in Field Arithmetic written by M. Fried and M. Jarden as following which is Proposition 13.4.6 in that monograph: Every field K has a regular extension F which is PAC ...
8
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1answer
193 views

Integral complete 4-partite graphs

For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$. Can such a graph be integral, i.e. have only integer eigenvalues? It is easy to see that the ...
4
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0answers
207 views

Action of the Picard Scheme of an Elliptic Fibration

Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
1
vote
1answer
233 views

Are there nontrivial rational solutions of $x^{n-m}=(1+t^m)/(1+t^n)$?

Let $n-m \ge 2$ be two fixed natural numbers. Are there any nontrivial rational solutions of the equation $$x^{n-m}=(1+t^m)/(1+t^n)$$ for $x$ and $t$? As particular cases the rational solutions of ...
9
votes
4answers
1k views

What is the smallest sphere whose surface includes 100 integer points?

Let $S(r)$ be the surface of the origin-centered sphere in $\mathbb{R}^3$. A point is an integer point if all its coordinates are integers. What is the smallest radius $r_n$ such that $S(r_n)$ ...
7
votes
1answer
546 views

Why study unirational and rational varieties?

I am new to the study of unirational and rational varieties, but I want to know the motivation for why mathematicians started to study these conditions. The reasons that I could list to study ...
2
votes
0answers
258 views

Number of rational points of a singular cubic surface over a finite field

I have a (geometrically irreducible) cubic surface defined over a finite field $F_q$ with three non-$F_q$-rational singularities (defined over the cubic extension of $F_q$). Counting the number of $...
2
votes
1answer
325 views

Counting algebraic points of bounded height

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\...
6
votes
1answer
206 views

rational points and a local perturbation of an elliptic curve

Let $E_{a,b}$ be an elliptic curve defined by the equation $y^{2}=x^3+ax+b$ where $a,b \in \mathbb{Q}$. Suppose that for $a=a_{0}$ and $b=b_{0}$ the rank of $E_{a_{0},b_{0}}(\mathbb{Q})=1$. question:...
15
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2answers
796 views

Determining the Mordell-Weil group of a universal elliptic curve

Let $K$ be a number field and let $K(a,b)$ be the field of rational functions with two indeterminates over $K$. Consider the elliptic curve $E$ over $K(a,b)$ defined by the Weierstrass equation \begin{...