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Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture 椭圆台球的有限性定理和动力学莫德尔-朗猜想的一个变体
1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-10-04 DOI: 10.1112/plms.12561
Pietro Corvaja, Umberto Zannier
{"title":"Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture","authors":"Pietro Corvaja, Umberto Zannier","doi":"10.1112/plms.12561","DOIUrl":"https://doi.org/10.1112/plms.12561","url":null,"abstract":"Abstract We offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in , there are only finitely many directions for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results (which are shown not to hold for general billiards) have their origin in ‘relative’ cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"48 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":"135547707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
L‐theory of C∗$C^*$‐algebras C$∗C^*$-gebras 的 L 理论
1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-10-03 DOI: 10.1112/plms.12564
Markus Land, Thomas Nikolaus, Marco Schlichting
{"title":"L‐theory of C∗$C^*$‐algebras","authors":"Markus Land, Thomas Nikolaus, Marco Schlichting","doi":"10.1112/plms.12564","DOIUrl":"https://doi.org/10.1112/plms.12564","url":null,"abstract":"Abstract We establish a formula for the L‐theory spectrum of real ‐algebras from which we deduce a presentation of the L‐groups in terms of the topological K‐groups, extending all previously known results of this kind. Along the way, we extend the integral comparison map obtained in previous work by the first two authors to real ‐algebras and interpret it using topological Grothendieck–Witt theory. Finally, we use our results to give an integral comparison between the Baum–Connes conjecture and the L‐theoretic Farrell–Jones conjecture, and discuss our comparison map in terms of the signature operator on oriented manifolds.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135744499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
2‐Selmer parity for hyperelliptic curves in quadratic extensions 二次扩展中超椭圆曲线的2‐Selmer宇称
1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-09-30 DOI: 10.1112/plms.12565
Adam Morgan
{"title":"2‐Selmer parity for hyperelliptic curves in quadratic extensions","authors":"Adam Morgan","doi":"10.1112/plms.12565","DOIUrl":"https://doi.org/10.1112/plms.12565","url":null,"abstract":"Abstract We study the 2‐parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels–Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen–Stoll. We establish the local formula in many instances and show that in remaining cases, it follows from standard global conjectures.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136247905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
An intrinsic approach to relative braid group symmetries on ı$imath$quantum groups 量子群上相对编织群对称性的一种内在方法
1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-09-29 DOI: 10.1112/plms.12562
Weiqiang Wang, Weinan Zhang
{"title":"An intrinsic approach to relative braid group symmetries on ı$imath$quantum groups","authors":"Weiqiang Wang, Weinan Zhang","doi":"10.1112/plms.12562","DOIUrl":"https://doi.org/10.1112/plms.12562","url":null,"abstract":"Abstract We initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135131835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Asymptotic enumeration of graphical regular representations 图正则表示的渐近枚举
1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-09-25 DOI: 10.1112/plms.12563
Binzhou Xia, Shasha Zheng
{"title":"Asymptotic enumeration of graphical regular representations","authors":"Binzhou Xia, Shasha Zheng","doi":"10.1112/plms.12563","DOIUrl":"https://doi.org/10.1112/plms.12563","url":null,"abstract":"Abstract We estimate the number of graphical regular representations (GRRs) of a given group with large enough order. As a consequence, we show that almost all finite Cayley graphs have full automorphism groups ‘as small as possible’. This confirms a conjecture of Babai–Godsil–Imrich–Lovász on the proportion of GRRs, as well as a conjecture of Xu on the proportion of normal Cayley graphs, among Cayley graphs of a given finite group.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
An identity in the Bethe subalgebra of C[Sn]$mathbb {C}[mathfrak {S}_n]$ C[Sn]$mathbb {C}[mathfrak {S}_n]$的Bethe子代数中的恒等式
1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-09-20 DOI: 10.1112/plms.12560
Kevin Purbhoo
{"title":"An identity in the Bethe subalgebra of C[Sn]$mathbb {C}[mathfrak {S}_n]$","authors":"Kevin Purbhoo","doi":"10.1112/plms.12560","DOIUrl":"https://doi.org/10.1112/plms.12560","url":null,"abstract":"Abstract As part of the proof of the Bethe ansatz conjecture for the Gaudin model for , Mukhin, Tarasov, and Varchenko described a correspondence between inverse Wronskians of polynomials and eigenspaces of the Gaudin Hamiltonians. Notably, this correspondence afforded the first proof of the Shapiro–Shapiro conjecture. In this paper, we give an identity in the group algebra of the symmetric group, which allows one to establish the correspondence directly, without using the Bethe ansatz.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136312761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The polynomials X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^{2} + (Y^3+Z^3)^2$ also capture their primes 多项式X2+（Y2+1）2$X^2+（Y^2+1）^2$和X2+（Y3+Z3）2$X ^｛2｝+（Y^3+Z^3）^2$也捕获它们的素数
IF 1.8 1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-08-19 DOI: 10.1112/plms.12557
Jori Merikoski
{"title":"The polynomials X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^{2} + (Y^3+Z^3)^2$ also capture their primes","authors":"Jori Merikoski","doi":"10.1112/plms.12557","DOIUrl":"https://doi.org/10.1112/plms.12557","url":null,"abstract":"We show that there are infinitely many primes of the form X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^2+(Y^3+Z^3)^2$ . This extends the work of Friedlander and Iwaniec showing that there are infinitely many primes of the form X2+Y4$X^2+Y^4$ . More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form. For the sequences X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^2+(Y^3+Z^3)^2$ , we establish Type II information that is too narrow for an aysmptotic formula, but we can use Harman's sieve method to produce a lower bound of the correct order of magnitude for primes of form X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^2+(Y^3+Z^3)^2$ . Estimating the Type II sums is reduced to a counting problem that is solved by using the Weil bound, where the arithmetic input is quite different from the work of Friedlander and Iwaniec for X2+Y4$X^2+Y^4$ . We also show that there are infinitely many primes p=X2+Y2$p=X^2+Y^2$ where Y$Y$ is represented by an incomplete norm form of degree k$k$ with k−1$k-1$ variables. For this, we require a Deligne‐type bound for correlations of hyper‐Kloosterman sums.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49657202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Issue Information 问题信息
IF 1.8 1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-08-01 DOI: 10.1112/plms.12456
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12456","DOIUrl":"https://doi.org/10.1112/plms.12456","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49023774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantitative bounds on vortex fluctuations in 2d$2d$ Coulomb gas and maximum of the integer-valued Gaussian free field 二维$二维$库仑气体涡旋波动的定量界和整数值高斯自由场最大值
IF 1.8 1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-07-23 DOI: 10.1112/plms.12551
Christophe Garban, Avelio Sepúlveda
{"title":"Quantitative bounds on vortex fluctuations in 2d$2d$ Coulomb gas and maximum of the integer-valued Gaussian free field","authors":"Christophe Garban, Avelio Sepúlveda","doi":"10.1112/plms.12551","DOIUrl":"https://doi.org/10.1112/plms.12551","url":null,"abstract":"In this paper, we study the influence of the vortices on the fluctuations of <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0003\" display=\"inline\" location=\"graphic/plms12551-math-0003.png\">\u0000<semantics>\u0000<mrow>\u0000<mn>2</mn>\u0000<mi>d</mi>\u0000</mrow>\u0000$2d$</annotation>\u0000</semantics></math> systems such as the Coulomb gas, the Villain model, or the integer-valued Gaussian free field (GFF). In the case of the <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0004\" display=\"inline\" location=\"graphic/plms12551-math-0004.png\">\u0000<semantics>\u0000<mrow>\u0000<mn>2</mn>\u0000<mi>d</mi>\u0000</mrow>\u0000$2d$</annotation>\u0000</semantics></math> Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin wave. We obtain the following quantitative upper bound on the two-point correlation in <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0005\" display=\"inline\" location=\"graphic/plms12551-math-0005.png\">\u0000<semantics>\u0000<msup>\u0000<mi mathvariant=\"double-struck\">Z</mi>\u0000<mn>2</mn>\u0000</msup>\u0000$mathbb {Z}^2$</annotation>\u0000</semantics></math> when <math altimg=\"urn:x-wiley:00246115:media:plms12551:plms12551-math-0006\" display=\"inline\" location=\"graphic/plms12551-math-0006.png\">\u0000<semantics>\u0000<mrow>\u0000<mi>β</mi>\u0000<mo>&gt;</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000$beta &gt;1$</annotation>\u0000</semantics></math>","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"27 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
The 2‐fusion system of the Monster 怪物的2‐聚变系统
IF 1.8 1区 数学
Proceedings of the London Mathematical Society Pub Date : 2023-07-19 DOI: 10.1112/plms.12549
M. Aschbacher
{"title":"The 2‐fusion system of the Monster","authors":"M. Aschbacher","doi":"10.1112/plms.12549","DOIUrl":"https://doi.org/10.1112/plms.12549","url":null,"abstract":"This paper is part of an effort to determine a certain class of simple 2‐fusion systems, and to use that result to simplify the proof of the classification of the finite simple groups. The main theorem proves that the 2‐fusion system of the Monster is the unique simple system with a fully centralized involution whose centralizer is the fusion system of the universal covering group of the Baby Monster.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"127 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41830451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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