. The goal of the paper is to explain a harmonic map approach to two geometric problems related to the Torelli map. The first is related to the existence of totally geodesic submanifolds in the image of the Torelli map, and the second is on rigidity of representation of a lattice of a semi-simple Lie group in a mapping class group.
{"title":"Torelli Locus and Rigidity","authors":"Sai-Kee Yeung","doi":"10.1307/mmj/20217207","DOIUrl":"https://doi.org/10.1307/mmj/20217207","url":null,"abstract":". The goal of the paper is to explain a harmonic map approach to two geometric problems related to the Torelli map. The first is related to the existence of totally geodesic submanifolds in the image of the Torelli map, and the second is on rigidity of representation of a lattice of a semi-simple Lie group in a mapping class group.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"38 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79537536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies residual finiteness of lattices in the universal cover of PU(2 , 1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2 , 1) or a finite covering of it. First, we prove that certain lattices in the universal cover of PU(2 , 1) are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in PU(2 , 1) to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in PU(2 , 1).
{"title":"Residually Finite Lattices in PU(2,1)˜ and Fundamental Groups of Smooth Projective Surfaces","authors":"Matthew Stover, D. Toledo","doi":"10.1307/mmj/20217215","DOIUrl":"https://doi.org/10.1307/mmj/20217215","url":null,"abstract":"This paper studies residual finiteness of lattices in the universal cover of PU(2 , 1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2 , 1) or a finite covering of it. First, we prove that certain lattices in the universal cover of PU(2 , 1) are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in PU(2 , 1) to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in PU(2 , 1).","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"70 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86313954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We give a classification of special reductive groups over arbitrary fields that improves a theorem of M. Huruguen.
． 给出了任意域上特殊约化群的一个分类，改进了M. hurugen的一个定理。
{"title":"Classification of Special Reductive Groups","authors":"A. Merkurjev","doi":"10.1307/mmj/20207201","DOIUrl":"https://doi.org/10.1307/mmj/20207201","url":null,"abstract":". We give a classification of special reductive groups over arbitrary fields that improves a theorem of M. Huruguen.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"35 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75689730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Integral Models and Torsors of Inseparable Forms of Ga","authors":"I. Dolgachev","doi":"10.1307/mmj/20217211","DOIUrl":"https://doi.org/10.1307/mmj/20217211","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"52 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86561845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the action of the derived Hecke algebra in the setting of dihedral weight one forms and prove a conjecture of the secondand fourthnamed authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as the ideas of Merel and Lecouturier on higher Eisenstein elements.
{"title":"The Derived Hecke Algebra for Dihedral Weight One Forms","authors":"M. Harris, V. Rotger, Akshay Venkatesh","doi":"10.1307/mmj/20217221","DOIUrl":"https://doi.org/10.1307/mmj/20217221","url":null,"abstract":"We study the action of the derived Hecke algebra in the setting of dihedral weight one forms and prove a conjecture of the secondand fourthnamed authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as the ideas of Merel and Lecouturier on higher Eisenstein elements.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"40 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89027134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H(SL1(D), R/Z) is a cyclic p-group whose order is bounded from below by the number of p-power roots of unity in F , unless D is a quaternion algebra over Q2. In this paper we give an explicit upper bound for the order of H(SL1(D), R/Z) for p ≥ 5, and determine H(SL1(D), R/Z) precisely when F is cyclotomic, p ≥ 19 and the degree of D is not a power of p.
{"title":"On the Second Cohomology of the Norm One Group of a p-Adic Division Algebra","authors":"M. Ershov, T. Weigel","doi":"10.1307/mmj/20217210","DOIUrl":"https://doi.org/10.1307/mmj/20217210","url":null,"abstract":"Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H(SL1(D), R/Z) is a cyclic p-group whose order is bounded from below by the number of p-power roots of unity in F , unless D is a quaternion algebra over Q2. In this paper we give an explicit upper bound for the order of H(SL1(D), R/Z) for p ≥ 5, and determine H(SL1(D), R/Z) precisely when F is cyclotomic, p ≥ 19 and the degree of D is not a power of p.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"417 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76601573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Rigidity of Some Extensions of Domains","authors":"Dayan Liu, Xiaosong Sun","doi":"10.1307/mmj/20205957","DOIUrl":"https://doi.org/10.1307/mmj/20205957","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"56 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89560978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rees-like algebras have played a major role in settling the EisenbudGoto conjecture. This article concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.
{"title":"Canonical Modules and Class Groups of Rees-Like Algebras","authors":"P. Mantero, J. McCullough, L. Miller","doi":"10.1307/mmj/20205974","DOIUrl":"https://doi.org/10.1307/mmj/20205974","url":null,"abstract":"Rees-like algebras have played a major role in settling the EisenbudGoto conjecture. This article concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"2004 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82949288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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