{"title":"On groups of finite Prüfer rank II","authors":"B. Wehrfritz","doi":"10.4171/rsmup/151","DOIUrl":"https://doi.org/10.4171/rsmup/151","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"57 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139385991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this note is to review the complex collar neighbourhood theorem of our paper [6] and add an extra precision to its proof. This is motivated by the importance of the subject, which attracted the attention of several authors, (see e.g. [2, 3, 4]). The two main ingredients consist of a local extension (or realizability) result (see [5]) and an ingenious use of the Zorn lemma which, in the non-compact case, is a substitute for the bumping technique of [1]. Our result is the following.
{"title":"On a complex collar neighbourhood theorem","authors":"C. Hill, M. Nacinovich","doi":"10.4171/rsmup/144","DOIUrl":"https://doi.org/10.4171/rsmup/144","url":null,"abstract":"The purpose of this note is to review the complex collar neighbourhood theorem of our paper [6] and add an extra precision to its proof. This is motivated by the importance of the subject, which attracted the attention of several authors, (see e.g. [2, 3, 4]). The two main ingredients consist of a local extension (or realizability) result (see [5]) and an ingenious use of the Zorn lemma which, in the non-compact case, is a substitute for the bumping technique of [1]. Our result is the following.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"10 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139168752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a finite group, we prove that every maximal subgroup of G has prime index if and only if every maximal subgroup of G that contains the normalizer of some Sylow subgroup has prime index, which implies that the hypothesis in Huppert’s theorem and the hypothesis in Chen’s theorem are actually equivalent. Moreover, we prove that the hypothesis in a theorem of Shao and Beltrán and the hypothesis in a theorem of Li et al are also equivalent.
{"title":"A note on Huppert's theorem and Chen's theorem","authors":"Jiangtao Shi, Mengjiao Shan, Fanjie Xu","doi":"10.4171/rsmup/153","DOIUrl":"https://doi.org/10.4171/rsmup/153","url":null,"abstract":"Let G be a finite group, we prove that every maximal subgroup of G has prime index if and only if every maximal subgroup of G that contains the normalizer of some Sylow subgroup has prime index, which implies that the hypothesis in Huppert’s theorem and the hypothesis in Chen’s theorem are actually equivalent. Moreover, we prove that the hypothesis in a theorem of Shao and Beltrán and the hypothesis in a theorem of Li et al are also equivalent.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"30 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138593508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Moore determinant","authors":"J. Fresnel, Michel Matignon","doi":"10.4171/rsmup/142","DOIUrl":"https://doi.org/10.4171/rsmup/142","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"118 38","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138599523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In previous work the (cid:12)rst two authors studied the notion of transitivity with respect to cyclic subgroups for separable Abelian p -groups and modules over the ring of p -adic integers. Here we consider brie(cid:13)y how the notion can be used in the context of torsion-free Abelian groups and also look at the situation for non-separable p -groups and direct sums of in(cid:12)nite rank homocyclic p -groups.
{"title":"Cyclic subgroup transitivity for Abelian groups","authors":"B. Goldsmith, K. Gong, L. Strüngmann","doi":"10.4171/rsmup/143","DOIUrl":"https://doi.org/10.4171/rsmup/143","url":null,"abstract":". In previous work the (cid:12)rst two authors studied the notion of transitivity with respect to cyclic subgroups for separable Abelian p -groups and modules over the ring of p -adic integers. Here we consider brie(cid:13)y how the notion can be used in the context of torsion-free Abelian groups and also look at the situation for non-separable p -groups and direct sums of in(cid:12)nite rank homocyclic p -groups.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138598543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
If $X$ is a smooth projective variety over ${mathbb R}$, the Hodge ${mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of good reduction is harder and was first studied by Spiess in the case of products of elliptic curve and by me in general.
{"title":"Abelian surfaces and the non-Archimedean Hodge-$D$-conjecture – The semi-stable case","authors":"Ramesh Sreekantan","doi":"10.4171/rsmup/139","DOIUrl":"https://doi.org/10.4171/rsmup/139","url":null,"abstract":"If $X$ is a smooth projective variety over ${mathbb R}$, the Hodge ${mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of good reduction is harder and was first studied by Spiess in the case of products of elliptic curve and by me in general.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"23 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135589692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
– Let S be the special fibre of a Shimura variety of Hodge type of good reduction at a fixed place above p . We give a local approach to the construction of the zip period map for S which is used to define the Ekedahl-Oort strata of S . The method employed is p -adic and group-theoretic in nature.
{"title":"A local construction of zip period maps of Shimura varieties","authors":"Qijun Yan","doi":"10.4171/rsmup/137","DOIUrl":"https://doi.org/10.4171/rsmup/137","url":null,"abstract":"– Let S be the special fibre of a Shimura variety of Hodge type of good reduction at a fixed place above p . We give a local approach to the construction of the zip period map for S which is used to define the Ekedahl-Oort strata of S . The method employed is p -adic and group-theoretic in nature.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"115 18","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135724946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Refinement of Gautschi's harmonic mean inequality for the gamma function","authors":"Horst Alzer","doi":"10.4171/rsmup/152","DOIUrl":"https://doi.org/10.4171/rsmup/152","url":null,"abstract":"In 1974, W. Gautschi proved that $$ 1<frac{2}{1/Gamma(x) +1/Gamma(1/x)} quad textrm{for} quad 0<xneq 1. $$ Here, we present the following refinement: $$ 1<GammaBigl( frac{2}{x+1/x}Bigr)< frac{2}{1/Gamma(x) +1/Gamma(1/x)}, quad 0<xneq 1. $$","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we define the $m$-prismatic site and the $m$-$q$-crystalline site, which are higher-level analogs of the prismatic site and the $q$-crystalline site respectively. We prove a certain equivalence between the category of crystals on the $m$-prismatic site (resp. the $m$-$q$-crystalline site) and that on the prismatic site (resp. the $q$-crystalline site), which can be regarded as the prismatic (resp. the $q$-crystalline) analog of the Frobenius descent due to Berthelot and the Cartier transform due to Ogus–Vologodsky, Oyama and Xu. We also prove equivalence between the category of crystals on the $m$-prismatic site and that on the $(m-1)$-$q$-crystalline site.
{"title":"Prismatic and $q$-crystalline sites of higher level","authors":"Kimihiko Li","doi":"10.4171/rsmup/136","DOIUrl":"https://doi.org/10.4171/rsmup/136","url":null,"abstract":"In this article, we define the $m$-prismatic site and the $m$-$q$-crystalline site, which are higher-level analogs of the prismatic site and the $q$-crystalline site respectively. We prove a certain equivalence between the category of crystals on the $m$-prismatic site (resp. the $m$-$q$-crystalline site) and that on the prismatic site (resp. the $q$-crystalline site), which can be regarded as the prismatic (resp. the $q$-crystalline) analog of the Frobenius descent due to Berthelot and the Cartier transform due to Ogus–Vologodsky, Oyama and Xu. We also prove equivalence between the category of crystals on the $m$-prismatic site and that on the $(m-1)$-$q$-crystalline site.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $C subset mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic $0$. For a line $l subset mathbb{P}^3$, we consider the projection $pi_lcolon C rightarrow mathbb{P}^1$ with center $l$ and the extension of the function fields $pi_l^astcolon k(mathbb{P}^1) hookrightarrow k(C)$. A line $l$ is assumed to be cyclic for $C$, if the extension $k(C)/pi_l^*(k(mathbb{P}^1))$ is cyclic. A line $l$ is assumed to be non-skew, if $C cap l ne emptyset$, i.e., $deg pi_l < deg C = 6$. We investigate the number of non-skew cyclic lines for $C$. As main results, we explicitly give the equation of $C$ in the particular case in which $C$ has two cyclic trigonal morphisms; we prove that the number of cyclic lines with $deg pi_l =4$ is at most~$1$, and the number of cyclic lines with $deg pi_l =5$ is at most~$1$.
设$C subset mathbb{P}^3$为特征为$0$的代数闭域$k$上的属$4$的正则曲线。对于直线$l subset mathbb{P}^3$，我们考虑以$l$为中心的投影$pi_lcolon C rightarrow mathbb{P}^1$和函数域$pi_l^astcolon k(mathbb{P}^1) hookrightarrow k(C)$的扩展。如果扩展名$k(C)/pi_l^*(k(mathbb{P}^1))$是循环的，则假定$C$的行$l$是循环的。假设一条直线$l$是非倾斜的，如果$C cap l ne emptyset$，即$deg pi_l < deg C = 6$。我们研究了$C$的非斜循环线的数目。作为主要结果，我们明确给出了$C$具有两个循环三角态射的特殊情况下$C$的方程;证明了$deg pi_l =4$的循环线数最多为$1$, $deg pi_l =5$的循环线数最多为$1$。
{"title":"Galois lines for a canonical curve of genus 4, I: Non-skew cyclic lines","authors":"Jiryo Komeda, Takeshi Takahashi","doi":"10.4171/rsmup/140","DOIUrl":"https://doi.org/10.4171/rsmup/140","url":null,"abstract":"Let $C subset mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic $0$. For a line $l subset mathbb{P}^3$, we consider the projection $pi_lcolon C rightarrow mathbb{P}^1$ with center $l$ and the extension of the function fields $pi_l^astcolon k(mathbb{P}^1) hookrightarrow k(C)$. A line $l$ is assumed to be cyclic for $C$, if the extension $k(C)/pi_l^*(k(mathbb{P}^1))$ is cyclic. A line $l$ is assumed to be non-skew, if $C cap l ne emptyset$, i.e., $deg pi_l < deg C = 6$. We investigate the number of non-skew cyclic lines for $C$. As main results, we explicitly give the equation of $C$ in the particular case in which $C$ has two cyclic trigonal morphisms; we prove that the number of cyclic lines with $deg pi_l =4$ is at most~$1$, and the number of cyclic lines with $deg pi_l =5$ is at most~$1$.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135828838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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