JournalsprimsVol. 55, No. 2pp. 401–451

A pp-adic Analytic Approach to the Absolute Grothendieck Conjecture

  • Takahiro Murotani

    Kyoto University, Japan
A $p$-adic Analytic Approach to the Absolute Grothendieck Conjecture cover
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Abstract

Let KK be a field, GKG_K the absolute Galois group of KK, XX a hyperbolic curve over KK and π1(X)\pi_1(X) the \'{e}tale fundamental group of XX. The absolute Grothendieck conjecture in anabelian geometry asks the following question: Is it possible to recover XX group-theoretically, solely from π1(X)\pi_1(X) (not π1(X)GK\pi_1(X)\twoheadrightarrow G_K)? When KK is a pp-adic field (i.e., a finite extension of Qp\mathbb{Q}_p), this conjecture (called the pp-adic absolute Grothendieck conjecture) is unsolved. To approach this problem, we introduce a certain pp-adic analytic invariant defined by Serre (which we call ii-invariant). Then the absolute pp-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 ii-invariants of the sets of rational points of the curve and its coverings; (B) recovering the ii-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).

Cite this article

Takahiro Murotani, A pp-adic Analytic Approach to the Absolute Grothendieck Conjecture. Publ. Res. Inst. Math. Sci. 55 (2019), no. 2, pp. 401–451

DOI 10.4171/PRIMS/55-2-6