In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module X ∞ of the Pontrjagin dual of the fine Selmer group of an elliptic curve E defined over ℚ. We consider the Galois extension field K n E of ℚ generated by coordinates of all p n -torsion points of E, and introduce a quotient A n E of the p-Sylow subgroup of the ideal class group of K n E cut out by the modulo p n Galois representation E[p n ]. We describe the asymptotic behavior of A n E by using the Iwasawa module X ∞ . In particular, under certain conditions, we obtain an asymptotic formula as Iwasawa’s class number formula on the order of A n E by using Iwasawa’s invariants of X ∞ .
本文研究了理想类群的某些商与椭圆曲线E上定义的精细Selmer群的Pontrjagin对偶的环切Iwasawa模X∞之间的关系。考虑由E的所有pn -扭力点的坐标生成的π的伽罗瓦扩展域K n E，并引入由模p n伽罗瓦表示E分割的K n E的理想类群的p- sylow子群的商an E[p n]。利用Iwasawa模X∞描述了A n E的渐近行为。特别地，在一定条件下，利用X∞上的Iwasawa不变量，得到了A ~ E阶的Iwasawa类数公式的渐近公式。
{"title":"Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves","authors":"Toshiro Hiranouchi, Tatsuya Ohshita","doi":"10.5802/jtnb.1258","DOIUrl":"https://doi.org/10.5802/jtnb.1258","url":null,"abstract":"In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module X ∞ of the Pontrjagin dual of the fine Selmer group of an elliptic curve E defined over ℚ. We consider the Galois extension field K n E of ℚ generated by coordinates of all p n -torsion points of E, and introduce a quotient A n E of the p-Sylow subgroup of the ideal class group of K n E cut out by the modulo p n Galois representation E[p n ]. We describe the asymptotic behavior of A n E by using the Iwasawa module X ∞ . In particular, under certain conditions, we obtain an asymptotic formula as Iwasawa’s class number formula on the order of A n E by using Iwasawa’s invariants of X ∞ .","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294166","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":"Sato–Tate Distributions of Catalan Curves","authors":"Heidi Goodson","doi":"10.5802/jtnb.1238","DOIUrl":"https://doi.org/10.5802/jtnb.1238","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43569392","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}
Takashi Miura, Kazuaki Murakami, K. Okano, Rei Otsuki
For an odd prime number p, we study the number of generators of the unramified Iwasawa modules of the maximal multiple Zp-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg’s generalized conjecture holds.
{"title":"On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple ℤ p -Extensions Over Imaginary Quadratic Fields","authors":"Takashi Miura, Kazuaki Murakami, K. Okano, Rei Otsuki","doi":"10.5802/jtnb.1232","DOIUrl":"https://doi.org/10.5802/jtnb.1232","url":null,"abstract":"For an odd prime number p, we study the number of generators of the unramified Iwasawa modules of the maximal multiple Zp-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg’s generalized conjecture holds.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45907585","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":"An a priori bound for rational functions on the Berkovich projective line","authors":"Y. Okuyama","doi":"10.5802/jtnb.1224","DOIUrl":"https://doi.org/10.5802/jtnb.1224","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43460229","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":"Factoring Gleason polynomials modulo 2","authors":"Xavier Buff, W. Floyd, Sarah C. Koch, W. Parry","doi":"10.5802/jtnb.1228","DOIUrl":"https://doi.org/10.5802/jtnb.1228","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47457923","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":"Control Theorems for Fine Selmer Groups","authors":"Debanjana Kundu, M. Lim","doi":"10.5802/jtnb.1231","DOIUrl":"https://doi.org/10.5802/jtnb.1231","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48669689","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":"Upper bounds and spectrum for approximation exponents for subspaces of ℝ n ","authors":"Elio Joseph","doi":"10.5802/jtnb.1230","DOIUrl":"https://doi.org/10.5802/jtnb.1230","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48059140","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}
Résumé. Le critère de Nyman–Beurling, équivalent à l’hypothèse de Riemann (HR), est un problème d’approximation dans l’espace des fonctions de carré intégrable sur (0 , ∞ ), par des dilatations de facteurs θ k ∈ (0 , 1), k ≥ 1, de la fonction partie fractionnaire. Prendre les θ k alétoires génère de nouvelles structures et de nouveaux critères. L’un d’eux est une condition suffisante pour HR qui revient à
{"title":"Polynomial approximations in a generalized Nyman–Beurling criterion","authors":"F. Alouges, Sébastien Darses, Erwan Hillion","doi":"10.5802/jtnb.1227","DOIUrl":"https://doi.org/10.5802/jtnb.1227","url":null,"abstract":"Résumé. Le critère de Nyman–Beurling, équivalent à l’hypothèse de Riemann (HR), est un problème d’approximation dans l’espace des fonctions de carré intégrable sur (0 , ∞ ), par des dilatations de facteurs θ k ∈ (0 , 1), k ≥ 1, de la fonction partie fractionnaire. Prendre les θ k alétoires génère de nouvelles structures et de nouveaux critères. L’un d’eux est une condition suffisante pour HR qui revient à","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46979576","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":"A note on Quebbemann’s extremal lattices of rank 64","authors":"I. Shimada","doi":"10.5802/jtnb.1229","DOIUrl":"https://doi.org/10.5802/jtnb.1229","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46574632","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":"Local distinction, quadratic base change and automorphic induction for GL n ","authors":"N. Matringe","doi":"10.5802/jtnb.1233","DOIUrl":"https://doi.org/10.5802/jtnb.1233","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43790395","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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