arXiv subjects
Publications and source records attributed to Wojciech Porowski.
We show that for a hyperbolic curve X over an algebraically closed field k of characteristic zero and a finite set S of k-rational points of X, after replacing X by a finite étale cover, there exists a family of curves over X whose fibres over points in S have pairwise nonisogenous Jacobians. We also give an application of our results to a problem concerning Galois sections.
We consider reductions of Selmer sections of the étale homotopy sequence of a hyperbolic curve over a number field. We show that the conjugacy class of a noncuspidal Selmer section is uniquely determined by its reduction on a set of density one. Moreover, we show that a noncuspidal Selmer section reduces to cusps only on a set of density zero, at least after a finite field extension.
We consider sections of the étale homotopy exact sequence of a hyperbolic curve over a number field. We prove that two sections whose restrictions to decomposition groups are conjugate on a set of valuations of density one are globally conjugate, which establishes the local-global principle for the conjugacy classes of sections. In fact, we obtain this result as a corollary of a more general property concerning sections of the étale homotopy exact sequence, so-called finite covering property, which we prove as our main result.