Polynomial bound for the number of rational points on curves. (Borne polynomiale pour le nombre de points rationnels des courbes.) (French. English summary) Zbl 1247.14022

Rémond, Gaël

J. Théor. Nombres Bordx. 23, No. 1, 251-255 (2011).

Let \(k\) be a number field, with discriminant \(\Delta\). Let \(f\in k[X,Y]\) be a polynomial of degree \(D\). The main theorem of the paper states that the set of affine \(k\)-rational points of \(f(x,y)=0\) is either infinite or bounded by \[ \text{exp}\left(2^{3^{D^2}}[K:{\mathbb Q}]^4 \max (h_\infty(F), \log|\Delta|, 1)\right), \] where \(h_\infty(F)\) denotes the logarithm of the maximum archimedean absolute value of the coefficients of \(F\).
The paper uses a simple estimate of the number of points of \({\mathbb P}^n(\bar{\mathbb Q})\) of degree \(\leq D\) and of height \(\leq H\) and compares with a previous paper of the same author [Proc. Lond. Math. Soc. (3) 101, No. 3, 759–794 (2010; Zbl 1210.11073)], improving the term of the estimate concerning the height but losing (substantially) on the dependence on the degree.

Reviewer: Roberto Dvornicich (Pisa)

MSC:

14G05	Rational points
11G50	Heights

Keywords:

Curves over number fields; number of rational points

Citations:

Zbl 1210.11073

References:

[1]	S. David et P. Philippon, Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes. Comment. Math. Helv. 77 (2002), 639-700. · Zbl 1030.11026
[2]	G. Rémond, Décompte dans une conjecture de Lang. Invent. math. 142 (2000), 513-545. · Zbl 0972.11054
[3]	G. Rémond, Nombre de points rationnels des courbes. Proc. London Math. Soc. 101 (2010), 759-794. · Zbl 1210.11073