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

Abstract

In the present paper, 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 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.