Uniform bounds on the number of rational points of a family of curves of genus 2∗
نویسندگان
چکیده
We exhibit a genus–2 curve C defined over Q(T ) which admits two independent morphisms to a rank–1 elliptic curve defined over Q(T ). We describe completely the set of Q(T )–rational points of the curve C and obtain a uniform bound for the number of Q–rational points of a rational specialization Ct of the curve C for a certain (possibly infinite) set of values t ∈ Q. Furthermore, for this set of values t ∈ Q we describe completely the set of Q–rational points of the curve Ct. Finally we show how these results can be strengthened assuming a height conjecture of S. Lang.
منابع مشابه
Uniform bounds for the number of rational points of families of curves of genus 2
We construct an infinite family {Ca,b}a,b∈Q of curves of genus 2 defined over Q, with two independent morphisms to a family of elliptic curves {Ea,b}a,b∈Q. When any of these elliptic curves Ea,b has rank 1 over Q, we obtain (modulo a conjecture of S. Lang, proved for special cases) a uniform bound for the number of rational points of the curve Ca,b, and an algorithm which finds all the rational...
متن کاملOn the Number of Rational Iterated Pre-images of the Origin under Quadratic Dynamical Systems
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article “Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,” by the present authors and five others, it was shown that the number of rational pre-images of the origin is bounded as on...
متن کاملUniform Bounds for the Number of Rational Points on Curves of Small Mordell–weil Rank
Let X be a curve of genus g 2 over a number field F of degree d D ŒF W Q . The conjectural existence of a uniform bound N.g;d/ on the number #X.F / of F rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound Ntors; .g; d/...
متن کاملOn the elliptic curves of the form $ y^2=x^3-3px $
By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. There is no known algorithm for finding the rank of this group. This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
متن کاملMAXIMAL PRYM VARIETY AND MAXIMAL MORPHISM
We investigated maximal Prym varieties on finite fields by attaining their upper bounds on the number of rational points. This concept gave us a motivation for defining a generalized definition of maximal curves i.e. maximal morphisms. By MAGMA, we give some non-trivial examples of maximal morphisms that results in non-trivial examples of maximal Prym varieties.
متن کامل