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タイトル: | Anabelian Geometry for Henselian Discrete Valuation Fields with Quasi-finite Residues |
著者: | MINAMIDE, Arata TSUJIMURA, Shota |
キーワード: | 11S20 11S31 anabelian geometry local class field theory Henselian discrete valuation field quasi-finite field cyclotomic rigidity Neukirch-Uchida theorem |
発行日: | Jun-2023 |
出版者: | Research Institute for Mathematical Sciences, Kyoto University |
開始ページ: | 1 |
終了ページ: | 68 |
論文番号: | RIMS-1973 |
抄録: | 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. |
URI: | http://hdl.handle.net/2433/284397 |
関連リンク: | https://www.kurims.kyoto-u.ac.jp/preprint/index.html |
出現コレクション: | 数理解析研究所プレプリント |
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