Arithmetic of del Pezzo surfaces of degree 1.

摘要

We study the density of rational points on del Pezzo surfaces of degree 1 for the Zariski topology and the adelic topology. For a large class of these surfaces over Q , we show that the set of rational points is dense for the Zariski topology. We achieve our results by carefully studying variations of root numbers among the fibers of elliptic surfaces associated to del Pezzo surfaces of degree 1. Our results in this direction are conditional on the finiteness of Tate-Shafarevich groups for elliptic curves over Q .;We also explicitly study the Galois action on the geometric Picard group of del Pezzo surfaces of degree 1 of the form w2=z3+Ax6+By6 in the weighted projective space P k(1, 1, 2, 3), where k is a global field of characteristic not 2 or 3 and A, B ∈ k*. Over a number field, we exhibit an infinite family of minimal surfaces for which the rational points are not dense for the adelic topology; i.e., minimal surfaces that fail to satisfy weak approximation. These counterexamples are explained by a Brauer-Manin obstruction.