On the [formula omitted]-adic section conjecture

Abstract

We prove that the analog of the Grothendieck anabelian section conjecture for curves holds true over a p-adic local field, if the following two assertions hold true. First, every section of the arithmetic fundamental group of a hyperbolic curve over a p-adic local field has an algebraic cycle class. Second, given a section of the cuspidally abelian absolute Galois group of the function field of such a curve, the corresponding section of the maximal Z/pZ-elementary abelian quotient, and a neighborhood of the latter, then the index of the corresponding curve is prime-to-p.