Pub Date : 2023-10-13DOI: 10.1017/s0305004123000592
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"PSP volume 175 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000592","DOIUrl":"https://doi.org/10.1017/s0305004123000592","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135858375","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}
Pub Date : 2023-10-13DOI: 10.1017/s0305004123000609
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"PSP volume 175 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000609","DOIUrl":"https://doi.org/10.1017/s0305004123000609","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135858175","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}
Pub Date : 2023-10-11DOI: 10.1017/s0305004123000579
Dessislava H. Kochloukova, Jone Lopez de Gamiz Zearra
Abstract We generalise some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if S is a full subdirect product of type $FP_s(mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of S to the direct product of any s of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to m . In particular, this gives the values of the analytic $L^2$ -Betti numbers of these groups in the respective dimensions.
{"title":"On subdirect products of type <i>FP<sub>n</sub></i> of limit groups over Droms RAAGs","authors":"Dessislava H. Kochloukova, Jone Lopez de Gamiz Zearra","doi":"10.1017/s0305004123000579","DOIUrl":"https://doi.org/10.1017/s0305004123000579","url":null,"abstract":"Abstract We generalise some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if S is a full subdirect product of type $FP_s(mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of S to the direct product of any s of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to m . In particular, this gives the values of the analytic $L^2$ -Betti numbers of these groups in the respective dimensions.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136058233","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}
Pub Date : 2023-10-10DOI: 10.1017/s0305004123000518
Yoav Len, Martin Ulirsch, Dmitry Zakharov
Abstract Let $mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $mathfrak{A}$ -covers of a tropical curve $Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $Gamma$ . We give a realisability criterion for harmonic $mathfrak{A}$ -covers by patching local monodromy data in an extended homology group on $Gamma$ . As an explicit example, we work out the case $mathfrak{A}=mathbb{Z}/pmathbb{Z}$ and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.
{"title":"Abelian tropical covers","authors":"Yoav Len, Martin Ulirsch, Dmitry Zakharov","doi":"10.1017/s0305004123000518","DOIUrl":"https://doi.org/10.1017/s0305004123000518","url":null,"abstract":"Abstract Let $mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $mathfrak{A}$ -covers of a tropical curve $Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $Gamma$ . We give a realisability criterion for harmonic $mathfrak{A}$ -covers by patching local monodromy data in an extended homology group on $Gamma$ . As an explicit example, we work out the case $mathfrak{A}=mathbb{Z}/pmathbb{Z}$ and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"51 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":"136254855","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}
Pub Date : 2023-10-09DOI: 10.1017/s0305004123000555
BACHIR BEKKA
Abstract Let $G= Nrtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H . The Bohr compactification ${rm Bohr}(G)$ and the profinite completion ${rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 rtimes {rm Bohr}(H)$ and $Q_2 rtimes {rm Prof}(H)$ for appropriate quotients $Q_1$ of ${rm Bohr}(N)$ and $Q_2$ of ${rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N . In the case where N is abelian, we have ${rm Bohr}(G)cong A rtimes {rm Bohr}(H)$ and ${rm Prof}(G)cong B rtimes {rm Prof}(H),$ where A (respectively B ) is the dual group of the group of unitary characters of N with finite H -orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= Lambdawr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${rm Bohr}(Lambdawr H)$ is isomorphic to ${rm Bohr}(Lambda^{rm Ab}wr H)$ and ${rm Prof}(Lambdawr H)$ is isomorphic to ${rm Prof}(Lambda^{rm Ab} wr H),$ where $Lambda^{rm Ab}=Lambda/ [Lambda, Lambda]$ is the abelianisation of $Lambda.$ As examples, we compute ${rm Bohr}(G)$ and ${rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.
{"title":"On Bohr compactifications and profinite completions of group extensions","authors":"BACHIR BEKKA","doi":"10.1017/s0305004123000555","DOIUrl":"https://doi.org/10.1017/s0305004123000555","url":null,"abstract":"Abstract Let $G= Nrtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H . The Bohr compactification ${rm Bohr}(G)$ and the profinite completion ${rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 rtimes {rm Bohr}(H)$ and $Q_2 rtimes {rm Prof}(H)$ for appropriate quotients $Q_1$ of ${rm Bohr}(N)$ and $Q_2$ of ${rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N . In the case where N is abelian, we have ${rm Bohr}(G)cong A rtimes {rm Bohr}(H)$ and ${rm Prof}(G)cong B rtimes {rm Prof}(H),$ where A (respectively B ) is the dual group of the group of unitary characters of N with finite H -orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= Lambdawr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${rm Bohr}(Lambdawr H)$ is isomorphic to ${rm Bohr}(Lambda^{rm Ab}wr H)$ and ${rm Prof}(Lambdawr H)$ is isomorphic to ${rm Prof}(Lambda^{rm Ab} wr H),$ where $Lambda^{rm Ab}=Lambda/ [Lambda, Lambda]$ is the abelianisation of $Lambda.$ As examples, we compute ${rm Bohr}(G)$ and ${rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135094897","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}
Pub Date : 2023-10-06DOI: 10.1017/s0305004123000567
DAWEI CHEN
Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k -differentials of infinite area are affine varieties for all k . Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.
{"title":"Nonvarying, affine and extremal geometry of strata of differentials","authors":"DAWEI CHEN","doi":"10.1017/s0305004123000567","DOIUrl":"https://doi.org/10.1017/s0305004123000567","url":null,"abstract":"Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k -differentials of infinite area are affine varieties for all k . Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"214 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351495","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}
Pub Date : 2023-10-06DOI: 10.1017/s0305004123000464
O. N. SILVA
Abstract We consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(mathbb{C}^2,0)$ to $(mathbb{C}^3,0)$ . We describe the embedded topological type of a generic hyperplane section of $f(mathbb{C}^2)$ , denoted by $gamma_f$ , in terms of the weights and degrees of f . As a consequence, a necessary condition for a corank 1 finitely determined map germ $g,{:},(mathbb{C}^2,0)rightarrow (mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.
{"title":"On the topology of the transversal slice of a quasi-homogeneous map germ","authors":"O. N. SILVA","doi":"10.1017/s0305004123000464","DOIUrl":"https://doi.org/10.1017/s0305004123000464","url":null,"abstract":"Abstract We consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(mathbb{C}^2,0)$ to $(mathbb{C}^3,0)$ . We describe the embedded topological type of a generic hyperplane section of $f(mathbb{C}^2)$ , denoted by $gamma_f$ , in terms of the weights and degrees of f . As a consequence, a necessary condition for a corank 1 finitely determined map germ $g,{:},(mathbb{C}^2,0)rightarrow (mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351398","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}
Pub Date : 2023-10-05DOI: 10.1017/s0305004123000543
ATHANASIOS SOURMELIDIS, JÖRN STEUDING
Abstract This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function $zeta$ . Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $zeta(sigma+it)$ for real t where $sigmain(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $zeta(sigma+it)$ when $sigma>1/2$ and we show that there is a connection with the zeros of $zeta'(sigma+it)$ . Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.
{"title":"Spirals of Riemann’s Zeta-Function — Curvature, Denseness and Universality","authors":"ATHANASIOS SOURMELIDIS, JÖRN STEUDING","doi":"10.1017/s0305004123000543","DOIUrl":"https://doi.org/10.1017/s0305004123000543","url":null,"abstract":"Abstract This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function $zeta$ . Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $zeta(sigma+it)$ for real t where $sigmain(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $zeta(sigma+it)$ when $sigma>1/2$ and we show that there is a connection with the zeros of $zeta'(sigma+it)$ . Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"33 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":"134975868","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}
Pub Date : 2023-10-05DOI: 10.1017/s0305004123000488
HUNG M. BUI, KYLE PRATT, ALEXANDRU ZAHARESCU
Abstract Erdős, Graham and Selfridge considered, for each positive integer n , the least value of $t_n$ so that the integers $n+1, n+2, dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$ , under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$ , and in the process solve Granville’s problem unconditionally.
{"title":"A problem of Erdős–Graham–Granville–Selfridge on integral points on hyperelliptic curves","authors":"HUNG M. BUI, KYLE PRATT, ALEXANDRU ZAHARESCU","doi":"10.1017/s0305004123000488","DOIUrl":"https://doi.org/10.1017/s0305004123000488","url":null,"abstract":"Abstract Erdős, Graham and Selfridge considered, for each positive integer n , the least value of $t_n$ so that the integers $n+1, n+2, dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$ , under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$ , and in the process solve Granville’s problem unconditionally.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"100 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":"134976029","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}
Pub Date : 2023-10-04DOI: 10.1017/s0305004123000531
MENG FAI LIM
Abstract This paper is concerned with the study of the fine Selmer group of an abelian variety over a $mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $mathbb{Z}_{p}[[ Gamma ]]$ , where $Gamma$ is the Galois group of the $mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $mathbb{Z}_{p}$ -extension, then it holds for almost all $mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p -adic Lie extension.
{"title":"Structure of fine Selmer groups over -extensions","authors":"MENG FAI LIM","doi":"10.1017/s0305004123000531","DOIUrl":"https://doi.org/10.1017/s0305004123000531","url":null,"abstract":"Abstract This paper is concerned with the study of the fine Selmer group of an abelian variety over a $mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $mathbb{Z}_{p}[[ Gamma ]]$ , where $Gamma$ is the Galois group of the $mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $mathbb{Z}_{p}$ -extension, then it holds for almost all $mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p -adic Lie extension.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591867","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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