A -adic Analytic Approach to the Absolute Grothendieck Conjecture
Takahiro Murotani
Kyoto University, Japan
![A $p$-adic Analytic Approach to the Absolute Grothendieck Conjecture cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-prims-volume-55-issue-2.png&w=3840&q=90)
Abstract
Let be a field, the absolute Galois group of , a hyperbolic curve over and the \'{e}tale fundamental group of . The absolute Grothendieck conjecture in anabelian geometry asks the following question: Is it possible to recover group-theoretically, solely from (not )? When is a -adic field (i.e., a finite extension of ), this conjecture (called the -adic absolute Grothendieck conjecture) is unsolved. To approach this problem, we introduce a certain -adic analytic invariant defined by Serre (which we call -invariant). Then the absolute -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 -invariants of the sets of rational points of the curve and its coverings; (B) recovering the -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 -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