Questions tagged [rational-points]
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216 questions
7
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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 ...
user39380
5
votes
1
answer
717
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 ...
- 51
0
votes
1
answer
371
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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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4
votes
0
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129
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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 ...
- 1,856
10
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4
answers
1k
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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 ...
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2
votes
1
answer
326
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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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3
votes
1
answer
256
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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. ...
- 913
3
votes
1
answer
173
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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 ...
user58604
6
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2
answers
297
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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\}...
- 13.6k
4
votes
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....
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1
vote
1
answer
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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 ...
- 8,998
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votes
3
answers
3k
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Integer-distance sets
Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...
- 151k
22
votes
5
answers
7k
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Rational points on a sphere in $\mathbb{R}^d$
Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.
Q1.
Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
- 151k
10
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4
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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)$ ...
- 151k
0
votes
1
answer
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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 $ ...
- 923
8
votes
1
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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 ...
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20
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5
answers
2k
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Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).
Contrary to the case of Fermat, ...
- 7,722
3
votes
1
answer
395
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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:
\...
- 31
20
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3
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what is the maximum number of rational points of a curve of genus 2 over the rationals
Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...
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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 $...
user168611
0
votes
1
answer
131
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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\...
- 25.4k
12
votes
0
answers
676
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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 ...
- 913
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265
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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 ...
- 7,182
4
votes
0
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195
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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 ...
- 2,224
11
votes
5
answers
4k
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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 ...
- 1,613
17
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1
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349
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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 ...
- 293
1
vote
0
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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' $. ...
- 923
4
votes
1
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415
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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 $\...
- 913
2
votes
2
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541
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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 ...
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20
votes
2
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2k
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Rational points on the "quintic circle" $x^5 + y^5 = 7$
I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are ...
- 1,329
0
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0
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87
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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 ...
- 923
5
votes
0
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303
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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))$, ...
6
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0
answers
233
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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 ...
- 19.2k
3
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1
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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
...
- 25.4k
1
vote
1
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353
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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 ...
- 895
4
votes
1
answer
423
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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, ...
- 913
10
votes
0
answers
217
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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 ...
- 2,051
1
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0
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194
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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 ...
- 895
2
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1
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187
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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 $(...
- 25.4k
15
votes
2
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1k
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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{...
- 20.8k
5
votes
4
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444
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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)$...
- 151k
0
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1
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544
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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 ...
user157361
19
votes
2
answers
1k
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Identifying Ramanujan's integer solutions of $x^3+y^3+z^3=1$ among Elkies' rational solutions
In his Lost Notebook, Ramanujan exhibits infinitely many integer solutions to $x^3+y^3+z^3=1$. On his webpage (http://www.math.harvard.edu/~elkies/4cubes.html), Elkies determines all rational ...
- 6,397
2
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0
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198
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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 $$
and let $\overline{C}$ denote the projective closure of $C$. For ...
- 21
1
vote
0
answers
147
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Properties of pointless projective curves over finite fields?
Probably not research level, feel free to downvote.
We got construction of bounded degree projective curves
with no points over finite fields. This construction generalizes to higher dimension.
One of ...
- 25.4k
2
votes
1
answer
188
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Criterion for existence of integral points on an elliptic curve
Is there a criterion for the (presumably infinite) set of $D \in \mathbb{Z}\setminus \{0\}$ such that
$$Dy^2 = x^3-1728$$
has an integral point over $\mathbb{Q}$ with $y \neq 0$? I'd also be ...
- 479
7
votes
1
answer
218
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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 ...
- 353
5
votes
0
answers
184
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 $\...
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2
votes
1
answer
600
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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, ...
- 403
10
votes
1
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563
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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 ...
- 149k