Grothendieck Conjecture for Hyperbolic Curves over Finitely Generated Fields of Positive Characteristic via Compatibility of Cyclotomes

Let p be a prime number. In the present paper, from the viewpoint of the compatibility/rigidity of group-theoretic cyclotomes, we revisit the anabelian Grothendieck Conjecture for hyperbolic curves over finitely generated fields of characteristic p established by A. Tamagawa, J. Stix, and S. Mochizuki. Especially, we give an alternative proof of the Grothendieck Conjecture for nonisotrivial hyperbolic curves over finitely generated fields of characteristic p obtained by J. Stix. In fact, by applying relatively recent results in anabelian geometry for hyperbolic curves over finite fields developed by M. Saïdi and A. Tamagawa, we discuss the J. Stix's result in a certain generalized situation, i.e., the geometrically pro-Σ setting, where Σ denotes the complement of a finite set of prime numbers that contains p in the set of all prime numbers. Moreover, by combining with a theorem in birational anabelian geometry obtained by F. Pop, we prove an absolute version of the geometrically pro-Σ Grothendieck Conjecture for nonisotrivial hyperbolic curves over the perfections of finitely generated fields of characteristic p. On the other hand, in the present paper, we also establish certain isotriviality criteria for hyperbolic curves with respect to both l-adic Galois representations and pro-l outer Galois representations, where l is a prime number ≠ p. These isotriviality criteria may be applied to give an alternative proof of the J. Stix's result

On the Semi-absoluteness of Isomorphisms between the Pro-pp Arithmetic Fundamental Groups of Smooth Varieties

Let p be a prime number. In the present paper, we consider a certain pro-p analogue of the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over p-adic local fields [i.e., finite extensions of the field of p-adic numbers ℚp​] obtained by Mochizuki. This research was motivated by Higashiyama’s recent work on the pro-p analogue of the semi-absolute version of the Grothendieck conjecture for configuration spaces [of dimension ≥2] associated to hyperbolic curves over generalized sub-p-adic fields [i.e., subfields of finitely generated extensions of the completion of the maximal unramified extension of ℚp​]

Construction of Non-×μ-Indivisible TKND-AVKF-Fields

In an author's joint work with Hoshi and Mochizuki, we introduced the notion of TKND-AVKF- fields [concerning the divisible subgroups of the groups of rationalpoints of semi-abelian varieties] and obtained an anabelian Grothendieck Conjecture-type result for higher dimensional con guration spaces associated to hyperbolic curvesover TKND-AVKF- fields. On the other hand, every concrete example of TKND-AVKF-fields. that appears in this joint work is a ×μ-indivisible eld [i.e., a eld such that anydivisible element of the multiplicative group of the eld is a root of unity]. In the presentpaper, we construct new examples of TKND-AVKF-fields that are not ×μ-indivisible

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

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

Construction of Abundant Explicit Nongeometric Pro-p Galois Sections of Punctured Projective Lines

In the present paper, we construct abundant explicit nongeometric pro-p Galois sections of certain punctured projective lines. Moreover, we also obtain an application to the theory of Massey products

Anabelian Geometry for Henselian Discrete Valuation Fields with Quasi-finite Residues

Let p, l be prime numbers. In anabelian geometry for p-adic local fields [i.e., finite extension fields of the field of p-adic numbers], many topics have been discussed. In the present paper, we generalize two of the topics --discovered by S. Mochizuki-- to more general complete discrete valuation fields. One is the mono-anabelian reconstruction, under a certain indeterminacy, of the cyclotomic rigidity isomorphism between the usual cyclotome Ẑ(1) associated to a p-adic local field and the cyclotome constructed, in a purely group-theoretic way, from [the underlying topological group structure of] the absolute Galois group of the p-adic local field. The other is the Neukirch-Uchida-type result, i.e., the field-theoreticity of an outer isomorphism between the absolute Galois groups of p-adic local fields that preserves the respective ramification filtrations. For our generalizations, we first discuss l-local class field theory for Henselian discrete valuation fields with strongly l-quasi-finite residue fields [i.e., perfect fields such that the maximal pro-l quotients of the absolute Galois groups of their finite extension fields are isomorphic to Ẑl] of characteristic p via Artin-Tate's class formation. This theory enables us to reconstruct the l-cyclotomes from the absolute Galois groups of such fields. With regard to cyclotomic rigidity, under a certain assumption, we establish mono-anabelian group/monoid-theoretic reconstruction algorithms for cyclotomic rigidity isomorphisms associated to Henselian discrete valuation fields with quasi-finite residue fields [i.e., perfect residue fields whose absolute Galois groups are isomorphic to Ẑ]. As an application of the reconstructions of cyclotomic rigidity isomorphisms, we determine the structure of the groups of Galois-equivariant automorphisms of various algebraically completed multiplicative groups that arise from complete discrete valuation fields with quasi-finite residues. Moreover, as a byproduct of the argument applied in this determination [especially, in the positive characteristic case], we also determine, in a generalized situation, the structure of a certain indeterminacy “(Ind2)” that appears in S. Mochizuki's inter-universal Teichmüller theory. With regard to the Neukirch-Uchida-type result, by combining the reconstruction result of p-cyclotomes above [in the case where l = p] with a recent result due to T. Murotani, together with a computation concerning norm maps, we prove an analogous result for mixed characteristic complete discrete valuation fields whose residue fields are [strongly] p-quasi-finite and algebraic over the prime fields

On the Injectivity of the Homomorphisms from the Automorphism Groups of Fields to the Outer Automorphism Groups of the Absolute Galois Groups

In the present paper, we discuss the injectivity of the natural homomorphism from the automorphism group of a given field to the outer automorphism group of the associated absolute Galois group. We prove that this natural homomorphism is injective in the case where, for instance, the given field may be embedded into the field of fractions of some Noetherian local domain of mixed characteristic

Anabelian Group-theoretic Properties of the Pro-p Absolute Galois Groups of Henselian Discrete Valuation Fields

Let p be a prime number; K a Henselian discrete valuation field of characteristic 0 such that the residue field is an infinite field of characteristic p. Write GK for the absolute Galois group of K. In our previous papers, under the assumption that K contains a primitive p-th root of unity ζp, we proved that any almost pro-p-maximal quotient of GK satisfies certain “anabelian” group-theoretic properties called very elasticity and strong internal indecomposability. In the present paper, we generalize this result to the case where K does not necessarily contain ζp. Then, by applying this generalization, together with some facts concerning Hilber-tian fields, we prove the semiabsoluteness of isomorphisms between thepro-p etale fundamental groups of smooth varieties over certain classes offields of characteristic 0. Moreover, we observe that there are various sim-ilarities between the maximal pro-p quotient GpK of GK and non abelianfree pro-p groups. For instance, we verify that every topologically finitely generated closed subgroup of GpK is a free pro-p group. One of the key ingredients of our proofs is “Artin-Schreier theory in characteristic zero”introduced by MacKenzie and Whaples

A Note on Stable Reduction of Smooth Curves Whose Jacobians Admit Stable Reduction

P. Deligne and D. Mumford proved that, for a smooth curve over the field of fractions of a discrete valuation ring whose residue field is perfect, if the associated Jacobian has stable reduction over the discrete valuation ring, then the smooth curve has stable reduction over the discrete valuation ring. Recently, I. Nagamachi proved a similar result over a connected normal Noetherian scheme of dimension one. In the present paper, we prove a similar result over a Prüfer domain, i.e., a domain whose localization at each of the prime ideals is a valuation ring. Moreover, we also give a counter-example in a situation over a higher dimensional base case. More precisely, we construct an example of a smooth curve over the field of fractions of a complete strictly Henselian normal Noetherian local domain of equal characteristic zero such that the associated Jacobian has good reduction over the local domain, but the smooth curve does not have stable reduction over the local domain

Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves

In this paper, we prove that arbitrary hyperbolic curves over p-adic local fields admit resolution of nonsingularities [“RNS”]. This result may be regarded as a generalization of results concerning resolution of nonsingularities obtained by A. Tamagawa and E. Lepage. Moreover, by combining our RNS result with techniques from combinatorial anabelian geometry, we prove that an absolute version of the geometrically pro-Σ Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields, where Σ denotes a set of prime numbers of cardinality ≥ 2 that contains p, holds. This settles one of the major open questions in anabelian geometry. Furthermore, we prove --again by applying RNS and combinatorial anabelian geometry-- that the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature in fact coincide. As a corollary, we conclude that the metrized Grothendieck-Teichmüller group is commensurably terminal in the Grothendieck-Teichmüller group. This settles a longstanding open question in combinatorial anabelian geometry