Questions tagged [rational-points]

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3
votes
2answers
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
1answer
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
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0answers
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
1answer
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
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1answer
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
2answers
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
1answer
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
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1answer
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
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1answer
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
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0answers
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
1answer
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
1answer
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
1answer
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
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0answers
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
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1answer
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
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4answers
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
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1answer
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
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1answer
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
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1answer
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
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0answers
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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0answers
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
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1answer
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
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0answers
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
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1answer
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
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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
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
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0answers
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
1answer
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
1answer
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
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2answers
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
1answer
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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0answers
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
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4answers
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
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4answers
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
2answers
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
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1answer
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
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0answers
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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1answer
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 ...
0
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0answers
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
1answer
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
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1answer
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
1answer
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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0answers
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
2answers
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
2answers
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
1answer
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
0answers
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
5answers
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
0answers
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
2answers
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} + \...