Existence of Nongeometric Pro-$p$ Galois Sections of Hyperbolic Curves

Abstract

We construct a nongeometric pro-$p$ Galois section of a proper hyperbolic curve over a number field, as well as over a $p$-adic local field. 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 field which admits infinitely many conjugacy classes of pro-$p$ Galois sections.