# Tag Info

## Hot answers tagged rational-points

Accepted

### Estimating the size of solutions of a diophantine equation

This problem turned out to be much more interesting than I originally thought. Let me give my solution, which seems to be slightly different from (but essentially the same as) the solution in the ...
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### Estimating the size of solutions of a diophantine equation

This exact problem is the subject of the paper "An Unusual Cubic Representation Problem" by Andrew Bremner (ASU) and myself. It was published in Volume 43 (2014) of Annales Mathematicae et ...
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### Status of $x^3+y^3+z^3=6xyz$

(Collecting comments into a community wiki answer.) There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's ...

### Rational points on the "quintic circle" $x^5 + y^5 = 7$

In his 1825 paper, Lejeune Dirichlet proved that the equation $x^5 + y^5 = cz^5$ has no nontrivial solution with x and y coprime integers and z integer for a rather large class of integers c. His ...
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### Average height of rational points on a curve

First, may I change your notation a bit? Usually one uses $H(p/q)=\max\{|p|,|q|\}$ for the (multiplicative) height of a rational number, and $h(p/q)=\log H(p/q)$ is the logarithmic height. So I'll use ...
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### what is the maximum number of rational points of a curve of genus 2 over the rationals

Here is some more information. The curve that establishes the current record is obtained from a K3 surface $S$ that was found by Noam Elkies. $S$ is a double cover of ${\mathbb P}^2$ ramified above a ...
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### Find all rational solutions of this diophantine-equation?

The number of rational solutions to your equation is finite. In short: your equation defines a genus $3$ curve, as follows from a straightforward computation and an application of Riemann--Hurwitz; ...
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### Conics, rational points and probability

Problems of this type are considered by Serre in the paper: Serre - Spécialisation des éléments de $\mathrm{Br}_2(\mathbb{Q}(T_1,\ldots, T_n))$ The case relevant to you is Exemple 4. Here Serre ...
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### Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

A conjecture of Coleman asserts that only finitely many rings arise as the endomorphism ring of an abelian variety of given dimension defined over a number field of given degree. See  for an ...
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### Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

Here are some references: MR2077618 Gundersen, Gary G.; Tohge, Kazuya Entire and meromorphic solutions of $f^5+g^5+h^5=1.$ Symposium on Complex Differential and Functional Equations, 57–67, Univ. ...
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For any given prime number $p > 2$ the probability that there is no $p$-adic point is $\ge c/p$ for some constant $c > 0$ indpendent of $p$. Since the sum over $1/p$ diverges, this implies that ...