On pro-p anabelian geometry for hyperbolic curves of genus 0 over p-adic fields

Abstract

Let p be a prime number. In the present paper, we discuss the relative／absolute version of the geometrically pro-p anabelian Grothendieck Conjecture (RpGC／ApGC). In the relative setting, we prove RpGC for hyperbolic curves of genus 0 over subfields of mixed characteristic valuation fields of rank 1 of residue characteristic p whose value groups have no nontrivial p-divisible element. In particular, one may take the completion of arbitrary tame extension of a mixed characteristic Henselian discrete valuation field of residue characteristic p as a base field. In light of the condition on base fields, this result may be regarded as a partial generalization of S. Mochizuki's classical anabelian result, i.e., RpGC for arbitrary hyperbolic curves over subfields of finitely generated fields of the completion of the maximal unramified extension of ℚp. It appears to the author that this result suggests that much wider class of p-adic fields may be considered as base fields in anabelian geometry. In the absolute setting, under the preservation of decomposition subgroups, we prove ApGC for hyperbolic curves of genus 0 over mixed characteristic Henselian discrete valuation fields of residue characteristic p. This result may be regarded as the first absolute Grothendieck Conjecture-type result for hyperbolic curves in the pro-p setting. Moreover, by combining this ApGC-type result with combinatorial anabelian geometry, under certain assumptions on decomposition groups and dimensions, we prove ApGC for configuration spaces of arbitrary hyperbolic curves over unramified extensions of p-adic local fields or their completions. In light of the condition on the dimension of configuration spaces, this result may be regarded as a partial generalization of a K. Higashiyama's pro-p semi-absolute Grothendieck Conjecture-type result.