Counting rational points on del Pezzo surfaces with a conic bundle structure

摘要

1. Introduction. Let k be a number field. A del Pezzo surface X over k is a non-singular projective surface defined over k, with ample anticanonical divisor -K_x The degree of X is defined to be d = (-Kx)~2. In this paper we will be concerned with upper bounds for the number of k-rational points of bounded height on del Pezzo surfaces of small degree. The arithmetic of del Pezzo surfaces becomes harder to understand as d decreases. For d ? {2,3,4} they admit the following classical description.