— The main purposes of the present paper are to introduce the notion of a hyperbolic curvoid and to study the geometry of hyperbolic curvoids. A hyperbolic curvoid is defined to be a certain profinite group and may be considered to be “group-theoretic abstraction” of the notion of a hyperbolic curve from the viewpoint of anabelian geometry. One typical example of a hyperbolic curvoid is a profinite group isomorphic to the étale fundamental group of a hyperbolic curve either over a number field or over a mixed-characteristic nonarchimedean local field. The first part of the present paper centers around establishments of a construction of the “geometric subgroup” of hyperbolic curvoids and a construction of the “collection of cuspidal inertia subgroups” of hyperbolic curvoids. Moreover, we also consider respective analogues for hyperbolic curvoids of the theory of partial compactifications of hyperbolic curves and the theory of quotient orbicurves of hyperbolic curves by actions of finite groups.
{"title":"The Geometry of Hyperbolic Curvoids","authors":"Yuichiro Hoshi","doi":"10.4171/prims/59-1-1","DOIUrl":"https://doi.org/10.4171/prims/59-1-1","url":null,"abstract":"— The main purposes of the present paper are to introduce the notion of a hyperbolic curvoid and to study the geometry of hyperbolic curvoids. A hyperbolic curvoid is defined to be a certain profinite group and may be considered to be “group-theoretic abstraction” of the notion of a hyperbolic curve from the viewpoint of anabelian geometry. One typical example of a hyperbolic curvoid is a profinite group isomorphic to the étale fundamental group of a hyperbolic curve either over a number field or over a mixed-characteristic nonarchimedean local field. The first part of the present paper centers around establishments of a construction of the “geometric subgroup” of hyperbolic curvoids and a construction of the “collection of cuspidal inertia subgroups” of hyperbolic curvoids. Moreover, we also consider respective analogues for hyperbolic curvoids of the theory of partial compactifications of hyperbolic curves and the theory of quotient orbicurves of hyperbolic curves by actions of finite groups.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42531980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Affine Super Schur Duality","authors":"Y. Flicker","doi":"10.4171/prims/59-1-5","DOIUrl":"https://doi.org/10.4171/prims/59-1-5","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43167196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bigraded Lie Algebras Related to Multiple Zeta Values","authors":"Mohamad Maassarani","doi":"10.4171/prims/58-4-4","DOIUrl":"https://doi.org/10.4171/prims/58-4-4","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46120349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boundedness of Operators and Inequalities on Morrey–Banach Spaces","authors":"K. Ho","doi":"10.4171/prims/58-3-4","DOIUrl":"https://doi.org/10.4171/prims/58-3-4","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44775118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Energy-Dependent Reflectionless Inverse Scattering","authors":"Yutaka Kamimura","doi":"10.4171/prims/58-2-5","DOIUrl":"https://doi.org/10.4171/prims/58-2-5","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47615806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Kobayashi Hyperbolicity of the Complements of Ample Divisors in Abelian Varieties","authors":"K. Yamanoi","doi":"10.4171/prims/58-2-2","DOIUrl":"https://doi.org/10.4171/prims/58-2-2","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41621753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sampei Hirose, T. Kawai, Shinji Sasaki, Yoshitsugu Takei
{"title":"On the Stokes Geometry of Perturbed Tangential Pearcey Systems","authors":"Sampei Hirose, T. Kawai, Shinji Sasaki, Yoshitsugu Takei","doi":"10.4171/prims/57-3-1","DOIUrl":"https://doi.org/10.4171/prims/57-3-1","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49410622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, which is the second in a series of four papers, we study theKummer theory surrounding the Hodge-Arakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the lth root of the] theta function at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point], for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th root of the] theta function at primes of bad reduction of EF in one Θ ±ellNF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in another Θ±ellNF-Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this “Θ-link”. One such new version is the Θ ×μ gaulink, which is similar to the Θ-link, but involves the theta values at l-torsion points, rather than the theta function itself. One important aspect of the constructions that underlie the Θ ×μ gau-link is the study of multiradiality properties, i.e., properties of the “arithmetic holomorphic structure” — or, more concretely, the ring/scheme structure — arising from one Θ±ellNF-Hodge theater that may be formulated in such a way as to make sense from the point of the arithmetic holomorphic structure of another Θ±ellNF-Hodge theater which is related to the original Θ±ellNF-Hodge theater by means of the [non-scheme-theoretic!] Θ ×μ gau-link. For instance, certain of the various rigidity properties of the étale theta function studied in an earlier paper by the author may be intepreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the Θ ×μ gau-link is the study of “conjugate synchronization” via the F ± l -symmetry of a Θ ±ellNF-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms — which are free of any conjugacy indeterminacies! — between copies of local absolute Galois groups at the various l-torsion points at which the theta function is evaluated. Conjugate synchronization plays an important role in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study of coricity properties of [i.e., the study of objects left invariant by] the Θ ×μ gau-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia gro
{"title":"Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation","authors":"S. Mochizuki","doi":"10.4171/PRIMS/57-1-2","DOIUrl":"https://doi.org/10.4171/PRIMS/57-1-2","url":null,"abstract":"In the present paper, which is the second in a series of four papers, we study theKummer theory surrounding the Hodge-Arakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the lth root of the] theta function at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point], for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th root of the] theta function at primes of bad reduction of EF in one Θ ±ellNF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in another Θ±ellNF-Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this “Θ-link”. One such new version is the Θ ×μ gaulink, which is similar to the Θ-link, but involves the theta values at l-torsion points, rather than the theta function itself. One important aspect of the constructions that underlie the Θ ×μ gau-link is the study of multiradiality properties, i.e., properties of the “arithmetic holomorphic structure” — or, more concretely, the ring/scheme structure — arising from one Θ±ellNF-Hodge theater that may be formulated in such a way as to make sense from the point of the arithmetic holomorphic structure of another Θ±ellNF-Hodge theater which is related to the original Θ±ellNF-Hodge theater by means of the [non-scheme-theoretic!] Θ ×μ gau-link. For instance, certain of the various rigidity properties of the étale theta function studied in an earlier paper by the author may be intepreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the Θ ×μ gau-link is the study of “conjugate synchronization” via the F ± l -symmetry of a Θ ±ellNF-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms — which are free of any conjugacy indeterminacies! — between copies of local absolute Galois groups at the various l-torsion points at which the theta function is evaluated. Conjugate synchronization plays an important role in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study of coricity properties of [i.e., the study of objects left invariant by] the Θ ×μ gau-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia gro","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":"57 1","pages":"209-401"},"PeriodicalIF":1.2,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48974996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper constitutes the third paper in a series of four papers and may be regarded as the culmination of the abstract conceptual portion of the theory developed in the series. In the present paper, we study the theory surrounding the log-theta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters. Here, we recall that Θ±ellNF-Hodge theaters were associated, in the first paper of the series, to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Each arrow of the log-theta-lattice corresponds to a certain gluing operation between the Θ±ellNF-Hodge theaters in the domain and codomain of the arrow. The horizontal arrows of the log-theta-lattice are defined as certain versions of the “Θ-link” that was constructed, in the second paper of the series, by applying the theory of HodgeArakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the l-th root of the] theta function at l-torsion points. In the present paper, we focus on the theory surrounding the log-link between Θ±ellNFHodge theaters. The log-link is obtained, roughly speaking, by applying, at each [say, for simplicity, nonarchimedean] valuation of the number field under consideration, the local p-adic logarithm. The significance of the log-link lies in the fact that it allows one to construct log-shells, i.e., roughly speaking, slightly adjusted forms of the image of the local units at the valuation under consideration via the local p-adic logarithm. The theory of log-shells was studied extensively in a previous paper by the author. The vertical arrows of the log-theta-lattice are given by the log-link. Consideration of various properties of the log-theta-lattice leads naturally to the establishment of multiradial algorithms for constructing “splitting monoids of logarithmic Gaussian procession monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. These logarithmic Gaussian procession monoids, or LGP-monoids, for short, may be thought of as the log-shell-theoretic versions of the Gaussian monoids that were studied in the second paper of the series. Finally, by applying these multiradial algorithms for splitting monoids of LGP-monoids, we obtain estimates for the log-volume of these LGP-monoids. Explicit computations of these estimates will be applied, in the fourth paper of the series, to derive various diophantine results. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI
{"title":"Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice","authors":"S. Mochizuki","doi":"10.4171/PRIMS/57-1-3","DOIUrl":"https://doi.org/10.4171/PRIMS/57-1-3","url":null,"abstract":"The present paper constitutes the third paper in a series of four papers and may be regarded as the culmination of the abstract conceptual portion of the theory developed in the series. In the present paper, we study the theory surrounding the log-theta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters. Here, we recall that Θ±ellNF-Hodge theaters were associated, in the first paper of the series, to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Each arrow of the log-theta-lattice corresponds to a certain gluing operation between the Θ±ellNF-Hodge theaters in the domain and codomain of the arrow. The horizontal arrows of the log-theta-lattice are defined as certain versions of the “Θ-link” that was constructed, in the second paper of the series, by applying the theory of HodgeArakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the l-th root of the] theta function at l-torsion points. In the present paper, we focus on the theory surrounding the log-link between Θ±ellNFHodge theaters. The log-link is obtained, roughly speaking, by applying, at each [say, for simplicity, nonarchimedean] valuation of the number field under consideration, the local p-adic logarithm. The significance of the log-link lies in the fact that it allows one to construct log-shells, i.e., roughly speaking, slightly adjusted forms of the image of the local units at the valuation under consideration via the local p-adic logarithm. The theory of log-shells was studied extensively in a previous paper by the author. The vertical arrows of the log-theta-lattice are given by the log-link. Consideration of various properties of the log-theta-lattice leads naturally to the establishment of multiradial algorithms for constructing “splitting monoids of logarithmic Gaussian procession monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. These logarithmic Gaussian procession monoids, or LGP-monoids, for short, may be thought of as the log-shell-theoretic versions of the Gaussian monoids that were studied in the second paper of the series. Finally, by applying these multiradial algorithms for splitting monoids of LGP-monoids, we obtain estimates for the log-volume of these LGP-monoids. Explicit computations of these estimates will be applied, in the fourth paper of the series, to derive various diophantine results. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":"57 1","pages":"403-626"},"PeriodicalIF":1.2,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46543314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logtheta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine — albeit from an extremely naive/non-expert point of view! — the foundational/settheoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.
{"title":"Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations","authors":"S. Mochizuki","doi":"10.4171/PRIMS/57-1-4","DOIUrl":"https://doi.org/10.4171/PRIMS/57-1-4","url":null,"abstract":"The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logtheta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine — albeit from an extremely naive/non-expert point of view! — the foundational/settheoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":"57 1","pages":"627-723"},"PeriodicalIF":1.2,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44918893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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