In this paper we study non-symplectic automorphisms of order 6 on $K3$ surfaces which are not purely. In particular we shall describe their fixed loci.
本文研究了$K3$曲面上非纯的6阶非辛自同构。特别地，我们将描述它们的固定位点。
{"title":"Non-purely non-symplectic automorphisms of order 6 on $K3$ surfaces","authors":"Nirai Shin-yashiki, Shingo Taki","doi":"10.3792/pjaa.97.012","DOIUrl":"https://doi.org/10.3792/pjaa.97.012","url":null,"abstract":"In this paper we study non-symplectic automorphisms of order 6 on $K3$ surfaces which are not purely. In particular we shall describe their fixed loci.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80274671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level N is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass forms.
{"title":"Shintani correspondence for Maass forms of level $N$ and prehomogeneous zeta functions","authors":"K. Sugiyama","doi":"10.3792/pjaa.98.008","DOIUrl":"https://doi.org/10.3792/pjaa.98.008","url":null,"abstract":"A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level N is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass forms.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76739587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary: We first announce our recent result on adjunction and inversion of adjunction. Then we clarify the relationship between our inversion of adjunction and Hacon’s inversion of adjunction for log canonical centers of arbitrary codimension.
{"title":"On inversion of adjunction","authors":"O. Fujino, K. Hashizume","doi":"10.3792/pjaa.98.003","DOIUrl":"https://doi.org/10.3792/pjaa.98.003","url":null,"abstract":"Summary: We first announce our recent result on adjunction and inversion of adjunction. Then we clarify the relationship between our inversion of adjunction and Hacon’s inversion of adjunction for log canonical centers of arbitrary codimension.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78626857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z × Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
{"title":"On elliptic curves induced by rational Diophantine quadruples","authors":"A. Dujella, G. Soydan","doi":"10.3792/pjaa.98.001","DOIUrl":"https://doi.org/10.3792/pjaa.98.001","url":null,"abstract":"In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z × Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80508092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: We report simple rigidity theorems for Euler products under deformations of Euler factors. Certain products of the Riemann zeta function are rigid in the sense that there exist no deformations which preserve the meromorphy on C .
{"title":"Rigidity of Euler products","authors":"S. Koyama, N. Kurokawa","doi":"10.3792/pjaa.97.016","DOIUrl":"https://doi.org/10.3792/pjaa.97.016","url":null,"abstract":": We report simple rigidity theorems for Euler products under deformations of Euler factors. Certain products of the Riemann zeta function are rigid in the sense that there exist no deformations which preserve the meromorphy on C .","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82520326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps are relative bicyclic biquadratic extensions. Regarding such towers as analogues of Z 2 -extensions, we examine the parity of 2-ideal class numbers along the towers with some examples.
{"title":"Iterated towers of number fields by a quadratic map defined over the Gaussian rationals","authors":"Yasushi Mizusawa, Kota Yamamoto","doi":"10.3792/PJAA.96.012","DOIUrl":"https://doi.org/10.3792/PJAA.96.012","url":null,"abstract":": An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps are relative bicyclic biquadratic extensions. Regarding such towers as analogues of Z 2 -extensions, we examine the parity of 2-ideal class numbers along the towers with some examples.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80712507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ${F_{n}}_{ngeq0}$ be the sequence of the Fibonacci numbers. The aim of this paper is to give explicit formulae for the infinite products [ prod_{n=1}^{infty}left( 1+frac{1}{F_{n}}right) ,qquadprod_{n=3}^{infty}left( 1-frac{1}{F_{n}}right) ] in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $mathbb{Q}$ of the above numbers by applying Bertrand's theorem on the algebraic independence of the values of the Jacobi theta functions.
{"title":"Algebraic independence of certain infinite products\u0000 involving the Fibonacci numbers","authors":"D. Duverney, Y. Tachiya","doi":"10.3792/PJAA.97.006","DOIUrl":"https://doi.org/10.3792/PJAA.97.006","url":null,"abstract":"Let ${F_{n}}_{ngeq0}$ be the sequence of the Fibonacci numbers. The aim of this paper is to give explicit formulae for the infinite products [ prod_{n=1}^{infty}left( 1+frac{1}{F_{n}}right) ,qquadprod_{n=3}^{infty}left( 1-frac{1}{F_{n}}right) ] in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $mathbb{Q}$ of the above numbers by applying Bertrand's theorem on the algebraic independence of the values of the Jacobi theta functions.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86919266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New results on slowly varying functions in the Zygmund sense","authors":"E. Omey, M. Cadena","doi":"10.3792/pjaa.96.009","DOIUrl":"https://doi.org/10.3792/pjaa.96.009","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76634563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $U_q'(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, we denote by $mathcal{F}_{mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $mathcal{D}$ to provide the functor $mathcal{F}_{mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $mathcal{C}^0_{mathfrak{g}}$, and show that all simple modules in $mathcal{C}^0_{mathfrak{g}}$ can be constructed as the heads of ordered tensor products of cuspidal modules.
{"title":"PBW theoretic approach to the module category of\u0000 quantum affine algebras","authors":"M. Kashiwara, Myungho Kim, Se-jin Oh, E. Park","doi":"10.3792/PJAA.97.007","DOIUrl":"https://doi.org/10.3792/PJAA.97.007","url":null,"abstract":"Let $U_q'(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, we denote by $mathcal{F}_{mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $mathcal{D}$ to provide the functor $mathcal{F}_{mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $mathcal{C}^0_{mathfrak{g}}$, and show that all simple modules in $mathcal{C}^0_{mathfrak{g}}$ can be constructed as the heads of ordered tensor products of cuspidal modules.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73887503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the diophantine equation F 1 þ 2F p 2 þ þ kF p k 1⁄4 F n, an equation on the weighted power terms of Fibonacci sequence. For the exponents p; q 2 f1; 2g the problem has already been solved in ad hoc ways using the properties of the summatory identities appear on the left-hand side of the equation. Here we suggest a uniform treatment for arbitrary positive integers p and q which works, in practice, for small values. We obtained all the solutions for p; q 10 by testing the new approach.
本文研究了关于斐波那契数列加权幂项的丢番图方程F 1 + 2F p 2 + kF p k 1 / 4 F n。对于指数p;q2f1;这个问题已经用特殊的方法解决了，利用等式左边的求和恒等式的性质。在这里，我们提出了一种对任意正整数p和q的统一处理方法，它在实践中适用于小数值。我们得到了p的所有解;q10通过测试新方法。
{"title":"On a Diophantine equation involving powers of Fibonacci numbers","authors":"K. Gueth, F. Luca, L. Szalay","doi":"10.3792/pjaa.96.007","DOIUrl":"https://doi.org/10.3792/pjaa.96.007","url":null,"abstract":"This paper deals with the diophantine equation F 1 þ 2F p 2 þ þ kF p k 1⁄4 F n, an equation on the weighted power terms of Fibonacci sequence. For the exponents p; q 2 f1; 2g the problem has already been solved in ad hoc ways using the properties of the summatory identities appear on the left-hand side of the equation. Here we suggest a uniform treatment for arbitrary positive integers p and q which works, in practice, for small values. We obtained all the solutions for p; q 10 by testing the new approach.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74880605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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