Let d be an integer, α = (-1 + √d) /2 if d ≡ 1 (mod 4), and α = √d otherwise. In this note we present elementary necessary and sufficient conditions for ℤ[α] to be a unique factorization domain. We then apply this result to produce sufficient conditions for ℤ[α] to be a unique factorization domain, in terms of prime-producing quadratic polynomials. We also apply this criterion to give an improvement of Rabinowitsch's result that provides necessary and sufficient conditions for the imaginary quadratic field K = ℚ(√1-4m), m ∈ ℕ, to have class number one. We also give two non-trivial applications to real quadratic number fields.
{"title":"NECESSARY AND SUFFICIENT CONDITIONS FOR UNIQUE FACTORIZATION IN ℤ[(-1 + √<i>d</i>)/2]","authors":"Víctor Julio RAMÍREZ VIÑAS","doi":"10.2206/kyushujm.77.121","DOIUrl":"https://doi.org/10.2206/kyushujm.77.121","url":null,"abstract":"Let d be an integer, α = (-1 + √d) /2 if d ≡ 1 (mod 4), and α = √d otherwise. In this note we present elementary necessary and sufficient conditions for ℤ[α] to be a unique factorization domain. We then apply this result to produce sufficient conditions for ℤ[α] to be a unique factorization domain, in terms of prime-producing quadratic polynomials. We also apply this criterion to give an improvement of Rabinowitsch's result that provides necessary and sufficient conditions for the imaginary quadratic field K = ℚ(√1-4m), m ∈ ℕ, to have class number one. We also give two non-trivial applications to real quadratic number fields.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367961","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 study prime tensor ideals in tensor abelian categories of quiver representations. Specifically, we classify the prime tensor ideals in the category of representations of zigzag quivers (with bounded path length) whose vertex set is the set of integers. We show that prime tensor ideals in these categories are in canonical bijection with prime ideals of a Boolean algebra, the power set of integers.
{"title":"PRIME TENSOR IDEALS IN ABELIAN CATEGORIES OF REPRESENTATIONS OF QUIVERS OF TYPE <i>A </i>","authors":"Shunsuke TADA","doi":"10.2206/kyushujm.77.159","DOIUrl":"https://doi.org/10.2206/kyushujm.77.159","url":null,"abstract":"We study prime tensor ideals in tensor abelian categories of quiver representations. Specifically, we classify the prime tensor ideals in the category of representations of zigzag quivers (with bounded path length) whose vertex set is the set of integers. We show that prime tensor ideals in these categories are in canonical bijection with prime ideals of a Boolean algebra, the power set of integers.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367969","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 work, using elementary tools we determine those Lattès maps which are at the same time Belyi maps by explicitly determining their ramification data. It turns out that in the generic case, i.e. when the automorphism group is Z/2Z, the corresponding family of Lattès maps are Belyi maps if the isogeny is multiplication by two or four. Elliptic curves with extra automorphisms also determine families of Belyi maps. We provide examples of some Belyi Lattès maps together with a formula for such maps which may be used to write Belyi maps of arbitrarily high degree. We conclude the paper with a discussion of the field of definition of such Belyi pairs.
{"title":"BELYI LATTÈS MAPS","authors":"Ayberk ZEYTIN","doi":"10.2206/kyushujm.77.221","DOIUrl":"https://doi.org/10.2206/kyushujm.77.221","url":null,"abstract":"In this work, using elementary tools we determine those Lattès maps which are at the same time Belyi maps by explicitly determining their ramification data. It turns out that in the generic case, i.e. when the automorphism group is Z/2Z, the corresponding family of Lattès maps are Belyi maps if the isogeny is multiplication by two or four. Elliptic curves with extra automorphisms also determine families of Belyi maps. We provide examples of some Belyi Lattès maps together with a formula for such maps which may be used to write Belyi maps of arbitrarily high degree. We conclude the paper with a discussion of the field of definition of such Belyi pairs.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367986","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}
The aims of this paper are expansions of the measure theory in Fesenko (Amer. Math. Soc, Transl. Ser. 2 219 (2006)) and Waller (New York J. Math. 25 (2019), 396-442). We define countably additive translation-invariant measures and integrals on product spaces of F and of GLd(F), and on a parabolic subgroup of GLd(F). Moreover, we will prove that measurable functions on these product spaces are repeatedly integrable.
本文的目的是对Fesenko (Amer)的测量理论进行扩展。数学。Soc, Transl。Ser. 2 219(2006))和Waller (New York J. Math. 25(2019)， 396-442)。定义了F和GLd(F)的积空间以及GLd(F)的抛物子群上的可数可加平移不变测度和积分。此外，我们将证明这些积空间上的可测函数是重复可积的。
{"title":"MEASURE THEORY ON GL<i><sub>d </sub></i>OVER A HIGHER-DIMENSIONAL LOCAL FIELD","authors":"Masaoki MORI","doi":"10.2206/kyushujm.77.271","DOIUrl":"https://doi.org/10.2206/kyushujm.77.271","url":null,"abstract":"The aims of this paper are expansions of the measure theory in Fesenko (Amer. Math. Soc, Transl. Ser. 2 219 (2006)) and Waller (New York J. Math. 25 (2019), 396-442). We define countably additive translation-invariant measures and integrals on product spaces of F and of GLd(F), and on a parabolic subgroup of GLd(F). Moreover, we will prove that measurable functions on these product spaces are repeatedly integrable.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367739","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 study the even periods of cusp forms on Γ0(p) associated to indefinite binary quadratic forms. In particular, we explicitly compute the even parts of the period polynomials of them and then show that they have rational even periods.
{"title":"CUSP FORMS WITH RATIONAL EVEN PERIODS","authors":"SoYoung CHOI","doi":"10.2206/kyushujm.77.141","DOIUrl":"https://doi.org/10.2206/kyushujm.77.141","url":null,"abstract":"We study the even periods of cusp forms on Γ0(p) associated to indefinite binary quadratic forms. In particular, we explicitly compute the even parts of the period polynomials of them and then show that they have rational even periods.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367761","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 formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily non-negative integers. Chinen discovered several families of formal weight enumerators and investigated the validity of the Riemann hypothesis analogue for them. In this paper, the zeta polynomial is computed for Chinen’s formal weight enumerators, and a simple criterion is given for the validity of the Riemann hypothesis analogue. real number q > 0 , q (cid:54)= 1. Moreover, if A σ ( x , y ) = ε A ( x , y ) holds for the
{"title":"ON THE RIEMANN HYPOTHESIS FOR A CERTAIN FAMILY OF FORMAL WEIGHT ENUMERATORS","authors":"Naoya Kaneko, Masakazu Yamagishi","doi":"10.2206/kyushujm.76.441","DOIUrl":"https://doi.org/10.2206/kyushujm.76.441","url":null,"abstract":". A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily non-negative integers. Chinen discovered several families of formal weight enumerators and investigated the validity of the Riemann hypothesis analogue for them. In this paper, the zeta polynomial is computed for Chinen’s formal weight enumerators, and a simple criterion is given for the validity of the Riemann hypothesis analogue. real number q > 0 , q (cid:54)= 1. Moreover, if A σ ( x , y ) = ε A ( x , y ) holds for the","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558505","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":"JACOBI-TRUDI FORMULA FOR THE HIGHER CAPELLI ELEMENTS OF CLASSICAL LIE ALGEBRAS","authors":"Shotaro Kawata, M. Noumi","doi":"10.2206/kyushujm.76.13","DOIUrl":"https://doi.org/10.2206/kyushujm.76.13","url":null,"abstract":"","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558877","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":"LINEAR COMBINATIONS OF HURWITZ ZETA-FUNCTIONS","authors":"R. Steuding, J. Steuding","doi":"10.2206/kyushujm.76.27","DOIUrl":"https://doi.org/10.2206/kyushujm.76.27","url":null,"abstract":"","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558422","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}
William Frendreiss, Jennifer Gao, Austin Lei, A. Woodall, Hui Xue, Daozhou Zhu
. For k < (cid:96) , let E k ( z ) and E (cid:96) ( z ) be Eisenstein series of weights k and (cid:96) , respectively, for SL 2 ( Z ) . We prove that between any two zeros of E k ( e i θ ) there is a zero of E (cid:96) ( e i θ ) on the interval π/ 2 < θ < 2 π/ 3.
． 对于k < (cid:96)，设ek (z)和E (cid:96) (z)分别为权k和权(cid:96)的Eisenstein级数，对于SL 2 (z)。证明了在π/ 2 < θ < 2 π/ 3区间上，在任意两个0之间存在一个0 E (cid:96) (E i θ)。
{"title":"A STIELTJES SEPARATION PROPERTY OF ZEROS OF EISENSTEIN SERIES","authors":"William Frendreiss, Jennifer Gao, Austin Lei, A. Woodall, Hui Xue, Daozhou Zhu","doi":"10.2206/kyushujm.76.407","DOIUrl":"https://doi.org/10.2206/kyushujm.76.407","url":null,"abstract":". For k < (cid:96) , let E k ( z ) and E (cid:96) ( z ) be Eisenstein series of weights k and (cid:96) , respectively, for SL 2 ( Z ) . We prove that between any two zeros of E k ( e i θ ) there is a zero of E (cid:96) ( e i θ ) on the interval π/ 2 < θ < 2 π/ 3.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558460","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}
The short-time asymptotic behavior of the transition density function of the diffusion process generated by the general Grushin operator will be investigated, by using its explicit expression in terms of expectation. Further the dependence on of the asymptotics will be seen.
{"title":"ON THE TRANSITION DENSITY FUNCTION OF THE DIFFUSION PROCESS GENERATED BY THE GRUSHIN OPERATOR","authors":"S. Taniguchi","doi":"10.2206/kyushujm.76.187","DOIUrl":"https://doi.org/10.2206/kyushujm.76.187","url":null,"abstract":"The short-time asymptotic behavior of the transition density function of the diffusion process generated by the general Grushin operator will be investigated, by using its explicit expression in terms of expectation. Further the dependence on of the asymptotics will be seen.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48729734","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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