Questions tagged [rational-points]
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229 questions
11
votes
5
answers
637
views
Looking for non-trivial parametric solutions to $(r^2+s^2)(r^2s^2+1)=t^2$
I am trying to find parametric families of rational solutions $(r, s, t) \in \mathbb{Q}^3$ for the Diophantine equation:
$$
(r^2 + s^2)(r^2 s^2 + 1) = t^2
$$
I consider the following solutions to be ...
- 111
4
votes
1
answer
256
views
Are there $a,b,c\in\mathbb Q$ so that $(ax^2+bx+c)^2=x^3+Ax+B$ has four solutions $x\in\mathbb Q$?
Consider an elliptic curve $E/\mathbb Q$ that has positive rank,
$$y^2=x^3+Ax+B.$$
Do there exist $a,b,c\in\mathbb Q$ such that there are distinct points $P_1,\ldots,P_4\in E(\mathbb Q)$ satisfying
$$...
- 427
1
vote
0
answers
252
views
Intersection of 6 quadratics define genus 1 curve with given 8 rational points
We found a genus 1 curve over the rationals with given 8 rational points and the defining equations are 6 quadratics using very crude opportunistic construction with linear algebra.
Q1 Why the genus ...
- 25.8k
1
vote
1
answer
127
views
Families of elliptic curves passing through three prescribed $x$-coordinates
Let take an elliptic curve which have a positive rank.
$$y^2=x^3+Ax+B$$
Dinote x-coordinates these rational points by
$$x_1,x_2,x_3,x_4,...$$
Now consider sliding triples of x-coordinates:
$$(x_n, x_{...
- 427
2
votes
0
answers
165
views
Does analytic rank always upper bound algebraic rank over function fields?
Let $A$ be an abelian variety over a global function field. Is the Mordell-Weil rank of $A$ known to be at most the analytic rank, i. e. the order at $s = 1$ of the $L$-function? This is stated in ...
- 1,146
0
votes
0
answers
108
views
Finiteness or Infiniteness of Solutions with $z = ax + b$ as a Perfect Square on an Elliptic Curve of Positive Rank
I am considering an elliptic curve of positive rank given by the equation
$$y^2 = x^3 + a x + b $$
where $a ≠ 0 $
Define
$$ z = y^2 - x^3 $$
so we have
$$z = a x + b$$
We know that the elliptic curve ...
- 427
3
votes
0
answers
380
views
Positive rank in the family of elliptic curves $y^2 = x^3 - a^2$ via Pell's equation
I am working on the elliptic curve
$$
E_k: y^2 = x^3 - a_k^2
$$
This elliptic curve is deeply connected with the Pell’s equation:
$$
a_k^2 - 3b_k^2 = 1
$$
We know that
$$
a_k + \sqrt{3} b_k = (2 + \...
- 427
0
votes
1
answer
247
views
Existence of 4-point rational loops on rank 0 elliptic curves?
I am studying elliptic curves and came across an interesting pattern.
For the elliptic curve:
$$
y^2 = x^3 - 219x + 1654.
$$
In this elliptic curve there are only $8$ solution couples $(x,y)$, the ...
- 427
1
vote
2
answers
341
views
Relationship between solutions of elliptic curve $y^2=x^3+ax+b$ and the equation $P^2=x-s$
I am studying elliptic curves of the form:
$$y^2 = x^3+ ax + b$$
Suppose this curve has positive rank, meaning it has infinitely many rational points $(x,y)$.
Now, I consider the case when $y=0$. ...
- 427
4
votes
1
answer
232
views
Is the curve $y^2=x^3-11n^2x\pm 14n^3$ related to the congruent number problem?
I would like to propose and ask about a conjecture involving a new family of elliptic curves that may be connected to the classical congruent number problem.
We know that a positive integer '$n$' is a ...
- 427
2
votes
2
answers
474
views
Does every geometrically connected scheme have a rational point?
Let $X$ be a connected algebraic scheme over a field $k$. It is well known that if $X$ has a rational point, then it is geometrically connected, i.e $X_{k^s}$ is connected.
Is the converse true? That ...
- 127
4
votes
1
answer
252
views
Rational points of Weierstrass equations over imperfect fields
Let $C$ be a singular projective curve over an imperfect field $K$ given by a Weierstrass equation. What is the structure of the group of non-singular rational points $C_\text{ns}(K)$?
Here is a ...
- 173
12
votes
2
answers
731
views
Does this elliptic curve over a cyclotomic tower have finitely many integral points?
$\mathbb{Q}(\zeta_{p^\infty})$, also written as $\mathbb{Q}(\mu_{p^\infty})$ or $\mathbb{Q}(p^\infty)$, denotes $\mathbb{Q}$ adjoined with the $p^{n}$th roots of unity for all $n$. It's the union of a ...
- 4,593
1
vote
0
answers
111
views
Number of symmetric matrices in a box of bounded determinant
Let $B$ and $T$ be positive real numbers. I'm interested in the following problem, which is about counting $2\times2$ symmetric matrices with bounded determinant and entries lying in a box:
Problem: ...
8
votes
1
answer
367
views
Average number of $\mathbb{F}_p$-points over twists of a variety
Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have:
Fact ...