A \(p\)-adic analytic approach to the absolute Grothendieck conjecture (Q2418850)

Summary: Let \(K\) be a field, \(G_K\) the absolute Galois group of \(K\), \(X\) a hyperbolic curve over \(K\) and \(\pi_1(X)\) the étale 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).