Uniform bounds for the number of rational points of families of curves of genus 2

PIERRICK GAUDRY; LEOPOLDO KULESZ; GUILLERMO MATERA; ERIC SCHOST

Buenos Aires

Conferencia; Jornadas Argentinas de Informática e Investigación Operativa; 2001

Sociedad Argentina de Informática e Investigación Operativa

We construct an infinite family {C_{a,b}}_{a,b in Q} of curves of genus 2 defined over Q, with two independent morphisms to a family of elliptic curves {E_{a,b}}_{a,b in Q}. When any of these elliptic curves E_{a,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 C_{a,b}, and an algorithm which finds all the rational points of C_{a,b}.