A $p$-adic Analytic Approach to the Absolute Grothendieck Conjecture

Abstract

Let K be a eld, G K the absolute Galois group of K, X a hyperbolic curve over K, and π 1 (X) the étale fundamental group of X. The absolute Grothendieck conjecture in anabelian geometry asks: Is it possible to recover X group-theoretically, solely from π 1 (X) (not π 1 (X) ↠ G K)? When K is a p-adic eld (i.e. a nite extension of Q p), this conjecture (called the p-adic absolute Grothendieck conjecture) is unsolved. To approach this problem, we introduce a certain p-adic analytic invariant dened by Serre (which we call i-invariant). Then, the absolute p-adic Grothendieck conjecture can be reduced to the following problems: (A) determining whether a proper hyperbolic curve admits a rational point from the data of i-invariants of the sets of rational points of the curve and its coverings; (B) recovering the i-invariant of the set of rational points of a proper hyperbolic curve group-theoretically. The main results of the present paper give a complete armative answer to (A) and a partial armative answer to (B).