# Questions tagged [rational-points]

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132
questions

**3**

votes

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72 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

289 views

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

The motiviation of this question is to look if there is such solution in rational number to the identity which montioned here, I have done many attempts using wolfram alpha to find such pairs of ...

**1**

vote

**0**answers

156 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

**1**answer

279 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

**1**answer

687 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

**2**answers

315 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**

votes

**1**answer

260 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, ...

**1**

vote

**1**answer

193 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**

votes

**1**answer

165 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**

votes

**0**answers

195 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

**1**answer

270 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

**1**answer

98 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 ...

**3**

votes

**1**answer

373 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 ...

**0**

votes

**0**answers

131 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 ...

**9**

votes

**1**answer

343 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

**4**answers

257 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

**1**answer

290 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**

votes

**1**answer

281 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

**1**answer

148 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 $\...

**1**

vote

**0**answers

83 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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**0**answers

267 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**

votes

**1**answer

189 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**

votes

**0**answers

179 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

**1**answer

230 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

**4**answers

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

**1**answer

475 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

**0**answers

232 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

**1**answer

318 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

**1**answer

205 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**

votes

**2**answers

756 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{...

**7**

votes

**1**answer

320 views

### Rational points on varieties over local fields

In his expanded lecture notes Rational points on varieties, Bjorn Poonen writes the following:
REMARK 2.5.3: There is an algorithm that, given a local field $k$ of characteristic $0$ and a $k$-...

**8**

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**0**answers

155 views

### Artin representations appearing in Mordell-Weil groups of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, and let $K$ be a Galois number field. The Galois group $G=\mathrm{Gal}(K/\mathbf{Q})$ acts on the Mordell-Weil group $E(K)$ and thus on the ...

**10**

votes

**4**answers

1k views

### Possible groups of K-rational points for elliptic curves over arbitrary fields

It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is ...

**8**

votes

**4**answers

306 views

### Scaling a set of reals to be nearly integers

A version of this question was previously asked on MSE. I'll mention progress below.
A geometric construction I'm exploring
leads to a set $R$ of $n$ positive real numbers, for example:
$$
R = \{ \pi,...

**3**

votes

**2**answers

401 views

### Does there exist a rational point on the elliptic curve: $y^2=x^3+6x^2+x$ ? If yes, how to find one? (relations to the 'rational distance problem') [closed]

The elliptic curve $y^2=x^3+6x^2+x$ is associated with the Rational Distance problem, which asks whether there exists a point in the plane, that is at rational distances from the four vertices of the ...

**1**

vote

**1**answer

249 views

### Rational points on open subsets of affine space [closed]

Let $k$ be an infinite field. Assume that the index of the algebraic closure $\bar{k}$ over $k$ is strictly greater than $2$. Let $U$ be a non-empty open subset of some affine space over $k$. Is it ...

**1**

vote

**0**answers

134 views

### Levi decompositions of k-rational points of linear algebraic groups

Let $k$ be a field with characteristic zero and $G$ be a (connected or not) linear algebraic group defined over $k$. We know that $G$ has a Levi decomposition i.e., $G=R_u(G)\rtimes L$, where $R_u(G)$ ...

**11**

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**1**answer

633 views

### Galois Representations and Rational Points

Suppose $X$ and $Y$ are two connected smooth projective varieties over $\mathbb{Q}$ (of the same dimension) that have the same $\ell$-adic Galois representations (up to semisimplification). What is ...

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**0**answers

106 views

### Existence of rational points on the image of a proper morphism

Let $K/F$ be a field extension, and let $X$ and $Y$ be affine varieties over $F$. (E.g. they are defined by polynomials over $F$.) Suppose $X$ contains $F$-points. Now view $X$ and $Y$ as $K$-...

**4**

votes

**1**answer

403 views

### Understanding Siegel's Theorem on integral points

Siegel's theorem states the following:
Let $C$ be a smooth projective curve over a number field $K$. Let $\tilde C\subset C$ be an open affine subvariety, and $i:\tilde C\hookrightarrow \mathbb{A}^...

**6**

votes

**1**answer

157 views

### If $X$ is a genus $g\geq 2$ curve over a number field $K$, then is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?

If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is ...

**7**

votes

**1**answer

417 views

### Does Chabauty-Coleman method give an algorithm for finding rational points?

Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see ...

**4**

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**0**answers

897 views

### Rational points on the unit circle

Is anything known about any of the following questions about rational points on the unit circle? By “double point” I mean an element of $2C$, where $C$ is the group of rational points on the unit ...

**3**

votes

**2**answers

187 views

### Existence of lattices whose circles have bounded number of points

For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as
$$\mathcal K(\Lambda) = \...

**8**

votes

**2**answers

642 views

### An elliptic curve for Ramanujan-type cubic identities?

Given the roots $x_i$ of the depressed cubic,
$$x^3+px+q=0$$
with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that,
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...

**1**

vote

**1**answer

214 views

### Heights of multiples of rational points on elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the ...

**6**

votes

**0**answers

94 views

### Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and ...

**11**

votes

**5**answers

2k views

### How much do I need to learn algebraic geometry to understand arithmetics over number fields

I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...

**2**

votes

**0**answers

97 views

### the least point on a variety over a finite field

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...

**130**

votes

**2**answers

50k views

### 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} + \...