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We begin by studying an abstract combinatorial analogue of the algebrogeometric 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 grouptheoretic 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 mixedcharacteristic 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 \u201cprop 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, \u201cpointtheoreticity implies geometricity\u201d; a \u201cnonconditional\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 \u201cpadically dense in a Galoiscompatible fashion\u201d. 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Topics in Absolute Anabelian Geometry II : Decomposition Groups and Endomorphisms
http://hdl.handle.net/2261/59041
27e992c6c3cd4c2ebccc887c9ad3c291
名前 / ファイル  ライセンス  アクション  

jms200201.pdf (572.7 kB)


Item type  紀要論文 / Departmental Bulletin Paper(1)  

公開日  20151215  
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 padic Grothendieck Conjecture  
主題Scheme  Other  
キーワード  
主題  padic 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  pointtheoreticity  
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 

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 threepart 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 algebrogeometric 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 grouptheoretic fashion, to this abstract combinatorial analogue; this leads to various anabelian results concerning configuration spaces. We then turn to the study of the absolute proΣ anabelian geometry of hyperbolic curves over mixedcharacteristic 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 “prop 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, “pointtheoreticity implies geometricity”; a “nonconditional” version of this result is then obtained for “procurves” obtained by removing from a proper curve some set of closed points which is “padically dense in a Galoiscompatible fashion”. Finally, we study, from an algorithmic point of view, the theory of Belyi and elliptic cuspidalizations, i.e., grouptheoretic 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. 171269, 発行日 20131120 

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  
20080326 