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We begin by studying an abstract combinatorial analogue of the algebro-geometric notion of a stable polycurve (i.e., a \u201csuccessive extension of families of stable curves\u201d) and showing that the \u201cgeometry of log divisors on stable polycurves\u201d may be extended, in a purely group-theoretic fashion, to this abstract combinatorial analogue; this leads to various anabelian results concerning configuration spaces. We then turn to the study of the absolute pro-\u03a3 anabelian geometry of hyperbolic curves over mixed-characteristic local fields, for \u03a3 a set of primes of cardinality \u2265 2 that contains the residue characteristic of the base field. In particular, we prove a certain \u201cpro-p resolution of nonsingularities\u201d type result, which implies a \u201cconditional\u201d anabelian result to the effect that the condition, on an isomorphism of arithmetic fundamental groups, of preservation of decomposition groups of \u201cmost\u201d closed points implies that the isomorphism arises from an isomorphism of schemes \u2014 i.e., in a word, \u201cpoint-theoreticity implies geometricity\u201d; a \u201cnon-conditional\u201d version of this result is then obtained for \u201cprocurves\u201d obtained by removing from a proper curve some set of closed points which is \u201cp-adically dense in a Galois-compatible fashion\u201d. 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Topics in Absolute Anabelian Geometry II : Decomposition Groups and Endomorphisms
http://hdl.handle.net/2261/59041
27e992c6-c3cd-4c2e-bccc-887c9ad3c291
| 名前 / ファイル | ライセンス | アクション | |
|---|---|---|---|
jms200201.pdf (572.7 kB)
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| Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
|---|---|---|---|---|---|---|
| 公開日 | 2015-12-15 | |||||
| タイトル | ||||||
| タイトル | Topics in Absolute Anabelian Geometry II : Decomposition Groups and Endomorphisms | |||||
| 言語 | ||||||
| 言語 | eng | |||||
| キーワード | ||||||
| 主題 | absolute anabelian geometry | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | hyperbolic curves | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | absolute p-adic Grothendieck Conjecture | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | p-adic Section Conjecture | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | configuration spaces | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | hidden endomorphisms | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | point-theoreticity | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | Belyi cuspidalization | |||||
| 主題Scheme | Other | |||||
| キーワード | ||||||
| 主題 | elliptic cuspidalization | |||||
| 主題Scheme | Other | |||||
| 資源タイプ | ||||||
| 資源 | http://purl.org/coar/resource_type/c_6501 | |||||
| タイプ | departmental bulletin paper | |||||
| 著者 |
Mochizuki, Shinichi
× Mochizuki, Shinichi |
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| 著者所属 | ||||||
| Research Institute for Mathematical Sciences, Kyoto University | ||||||
| 抄録 | ||||||
| 内容記述タイプ | Abstract | |||||
| 内容記述 | 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 “geometry 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. We 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 nonsingularities” type result, which implies a “conditional” anabelian result to the effect that the condition, on an isomorphism of arithmetic fundamental 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 “procurves” obtained by removing from a proper curve some set of closed points which is “p-adically dense in a Galois-compatible fashion”. Finally, we study, from an algorithmic point of view, the theory of Belyi and elliptic cuspidalizations, i.e., group-theoretic reconstruction algorithms for the arithmetic fundamental group of an open subscheme of a hyperbolic curve that arise from consideration of certain endomorphisms determined by Belyi maps and endomorphisms of elliptic curves. | |||||
| 書誌情報 |
Journal of mathematical sciences, the University of Tokyo 巻 20, 号 2, p. 171-269, 発行日 2013-11-20 |
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| ISSN | ||||||
| 収録物識別子タイプ | ISSN | |||||
| 収録物識別子 | 13405705 | |||||
| 書誌レコードID | ||||||
| 収録物識別子タイプ | NCID | |||||
| 収録物識別子 | AA11021653 | |||||
| 日本十進分類法 | ||||||
| 主題 | 415 | |||||
| 主題Scheme | NDC | |||||
| Mathematical Reviews Number | ||||||
| MR | ||||||
| Mathmatical Subject Classification | ||||||
| 14H30(MSC2010) | ||||||
| Mathmatical Subject Classification | ||||||
| 14H25(MSC2010) | ||||||
| 出版者 | ||||||
| 出版者 | Graduate School of Mathematical Sciences, The University of Tokyo | |||||
| 原稿受領日 | ||||||
| 2008-03-26 | ||||||
