arXiv subjects

Takeo Uramoto

Publications and source records attributed to Takeo Uramoto.

6 recordsLinked to original sources

arXiv2026

Deformation of the absolute Galois groups of number fields

The technical goal of this paper is to construct and study profinite monoids DG_K for number fields K such that (1) the unit group of DG_K is isomorphic to the absolute Galois group G_K; (2) the maximal abelian quotient of DG_K is isomorphic to the Deligne-Ribet monoid DR_K; (3) the idempotents of DG_K bijectively correspond to subsets of the set P_K of (finite) primes of K; (4) the maximal Galois groups G_{K, S} with restricted ramifications S all appear as maximal closed subgroups of DG_K at idempotents; (5) all maximal closed subgroups of DG_K at idempotents are of this form. We also discuss the relationship between DG_K and the p-typical Witt vectors of Borger and de Smit, where we will describe the fundamental monoid of the semi-galois category of Lambda-rings of Borger and de Smit in terms of fields of norms of the local field K_p in particular.

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arXiv2025

Absolute reconstruction of number fields from the Deligne-Ribet monoids

Following Cornelissen, Li, Marcolli, and Smit, this short paper proves that the field structure of a number field $K$ can be reconstructed from the pair $(DR_K, I_K)$ of the Deligne-Ribet monoid $DR_K$ and the submonoid $I_K$ of $DR_K$, when $K$ is the rational number field, or an imaginary quadratic field. The general-case reconstruction is also discussed, which is more abstract than the case of rational and imaginary quadratic fields.

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arXiv2024

Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions

This paper is a sequel to our previous work, where we proved the ``modularity theorem'' for algebraic Witt vectors over imaginary quadratic fields. This theorem states that, in the case of imaginary quadratic fields $K$, the algebraic Witt vectors over $K$ are precisely those generated by the modular vectors whose components are given by special values of deformation family of Fricke modular functions; arithmetically, this theorem implies certain congruences between special values of modular functions that are not necessarily galois conjugate. In order to take a closer look at this modularity theorem, the current paper extends it to the case of CM fields. The main results include (i) a construction of algebraic Witt vectors from special values of deformation family of Siegel modular functions on Siegel upper-half space given by ratios of theta functions, and (ii) a galois-theoretic characterization of which algebraic Witt vectors arise in this modular way, intending to exemplify a general galois-correspondence result which is also proved in this paper.

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arXiv2021

Semi-galois Categories III: Witt vectors by deformations of modular functions

Based on our previous work on an arithmetic analogue of Christol's theorem, this paper studies in more detail the structure of the lambda-ring $E_K = K \otimes W_{O_K}^a (O_{\bar{K}})$ of algebraic Witt vectors for number fields $K$. First developing general results concerning $E_K$, we apply them to the case when $K$ is an imaginary quadratic field. The main results include the "modularity theorem" for algebraic Witt vectors, which claims that certain deformation families $f: M_2(\widehat{\mathbb{Z}}) \times \mathfrak{H} \rightarrow \mathbb{C}$ of modular functions of finite level always define algebraic Witt vectors $\widehat{f}$ by their special values, and conversely, every algebraic Witt vector $ξ\in E_K$ is realized in this way, that is, $ξ= \widehat{f}$ for some deformation family $f: M_2(\widehat{\mathbb{Z}}) \times \mathfrak{H} \rightarrow \mathbb{C}$. This gives a rather explicit description of the lambda-ring $E_K$ for imaginary quadratic fields $K$, which is stated as the identity $E_K=M_K$ between the lambda-ring $E_K$ and the $K$-algebra $M_K$ of modular vectors $\widehat{f}$.

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arXiv2018

Semi-galois Categories II: An arithmetic analogue of Christol's theorem

In connection with our previous work on semi-galois categories, this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series over finite field is algebraic over the polynomial ring. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F_1) variant of Christol's theorem in the sense that it replaces the polynomial ring with the ring of integers of a number field and the ring of formal power series with the ring of Witt vectors. We also study some related problems.

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arXiv2017

Semi-galois Categories I: The Classical Eilenberg Variety Theory

This paper is an extended version of our proceedings paper announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspect on our work. Technically, this paper introduces and studies the class of semi-galois categories, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of Eilenberg's theory of varieties of regular languages--- a branch in formal language theory that has been developed since the mid 1960's and particularly concerns systematic classification of regular languages, finite monoids, and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on this results on semi-galois categories, we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi BM of profinite monoids M among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos E is equivalent to the classifying topos BM of some profinite monoid M if and only if E is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point. This topos-theoretic consideration is related to the logical and geometric problems concerning Eilenberg's theory that we addressed at LICS'16, which remain open in this paper.

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