Existence of Nongeometric Pro-<i>p</i> Galois Sections of Hyperbolic Curves

Abstract

We construct a nongeometric pro-p Galois section of a proper hyperbolic curve over a number field, as well as over a p-adic local field. This yields a negative answer to the pro-p version of the anabelian Grothendieck Section Conjecture. We also observe that there exists a proper hyperbolic curve over a number field which admits infinitely many conjugacy classes of pro-p Galois sections.