Showing papers by "Shinichi Mochizuki published in 2013"

Topics in absolute anabelian geometry ii: decomposition groups and endomorphisms

Abstract: The present paper, which forms the second part of a three-part series in which we study absolute anabelian geometry from an algorithmic point of view, focuses on the study of the closely related notions of decomposition groups and endomorphisms in this anabelian context. We begin by studying an abstract combinatorial analogue of the algebro-geometric notion of a stable polycurve (i.e., a "successive extension of families of stable curves") and showing that the "geome- try of log divisors on stable polycurves" may be extended, in a purely group-theoretic fashion, to this abstract combinatorial analogue; this leads to various anabelian results concerning configuration spaces .W e then turn to the study of the absolute pro-Σ anabelian geometry of hyperbolic curves over mixed-characteristic local fields, for Σ a set of primes of cardinality ≥ 2 that contains the residue characteristic of the base field. In particular, we prove a certain "pro-p resolution of nonsin- gularities" type result, which implies a "conditional" anabelian result to the effect that the condition, on an isomorphism of arithmetic fun- damental groups, of preservation of decomposition groups of "most" closed points implies that the isomorphism arises from an isomorphism of schemes — i.e., in a word, "point-theoreticity implies geometricity"; a "non-conditional" version of this result is then obtained for "pro- curves" obtained by removing from a proper curve some set of closed points which is "p-adically dense in a Galois-compatible fashion". Fi- nally, we study, from an algorithmic point of view, the theory of Belyi and elliptic cuspidalizations, i.e., group-theoretic reconstruction algo- rithms for the arithmetic fundamental group of an open subscheme of a hyperbolic curve that arise from consideration of certain endo- morphisms determined by Belyi maps and endomorphisms of elliptic curves.

Topics surrounding the combinatorial anabelian geometry of hyperbolic curves iii: tripods and tempered fundamental groups

Abstract: Let Σ be a subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinal- ity one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated con- figuration spaces over algebraically closed fields in which the primes of Σ are invertible. The focus of the present paper is on appli- cations of the theory developed in previous papers to the theory of tempered fundamental groups, in the style of Andre. These applications are motivated by the goal of surmounting two funda- mental technical difficulties that appear in previous work of Andre, namely: (a) the fact that the characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve that is given in earlier work of Andre is only proven for a quite limited class of hyperbolic curves, i.e., a class that is "far from generic" ;( b) the proof given in earlier work of Andre of a certain key injectivity result, which is of central importance in establishing the theory of a" p-adic local analogue" of the well-known "global" theory of the Grothendieck-Teichmuller group, contains a fundamental gap .I n the present paper, we surmount these technical difficulties by introduc- ing the notion of an "M-admissible" ,o r"metric-admissible", outer automorphism of the profinite geometric fundamental group of a p-adic hyperbolic curve. Roughly speaking, M-admissible outer automorphisms are outer automorphisms that are compatible with the data constituted by the indices at the various nodes of the spe- cial fiber of the p-adic curve under consideration. By combining this notion with combinatorial anabelian results and techniques de- veloped in earlier papers by the authors, together with the theory of cyclotomic synchronization (also developed in earlier papers by the authors), we obtain a generalization of Andre's characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve to the case of arbitrary hyperbolic curves (cf. (a)). Moreover, by applying the theory of local contractibility of p -adic analytic spacesdeveloped by Berkovich, we show that the techniques developed in the present and earlier papers by the authors allow one to relate the groups of M-admissible outer automorphisms treated in the present paper to the groups of outer automorphisms