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

Abstract

Let $K$ be a field, $G_K$ the absolute Galois group of $K$, $X$ a hyperbolic curve over $K$ and ${\pi }_1(X)$ the \'{e}tale fundamental group of $X$. The absolute Grothendieck conjecture in anabelian geometry asks the following question: Is it possible to recover $X$ group-theoretically, solely from ${\pi }_1(X)$ (not ${\pi }_1(X)\twoheadrightarrow G_K$)? When $K$ is a $p$-adic field (i.e., a finite extension of ${\mathbb{Q}}_p$), this conjecture (called the $p$-adic absolute Grothendieck conjecture) is unsolved. To approach this problem, we introduce a certain $p$-adic analytic invariant defined 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 the $i$-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 affirmative answer to (A) and a partial affirmative answer to (B).