We study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a field with coefficients in Z/ℓ$mathbb {Z}/ell$ . We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish.
研究了在系数为Z/ r $mathbb {Z}/ell$的域上绝对不可约光滑投影曲线的至少3阶Massey积的消失性。我们主要关注椭圆曲线，得到了三重Massey积不消失时的完整表征。
{"title":"Massey products and elliptic curves","authors":"F. Bleher, T. Chinburg, J. Gillibert","doi":"10.1112/plms.12541","DOIUrl":"https://doi.org/10.1112/plms.12541","url":null,"abstract":"We study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a field with coefficients in Z/ℓ$mathbb {Z}/ell$ . We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42657452","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}
We obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by the second largest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.
{"title":"Poincaré constant on manifolds with ends","authors":"A. Grigor’yan, Satoshi Ishiwata, L. Saloff‐Coste","doi":"10.1112/plms.12522","DOIUrl":"https://doi.org/10.1112/plms.12522","url":null,"abstract":"We obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by the second largest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47216874","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}
The figure‐eight knot exterior N0$N_0$ supports a natural DA (derived from Anosov) expanding attractor, with which Franks–Williams constructed the first example of non‐transitive Anosov flow. This flow lies in a 3‐manifold M0$M_0$ which is the double of N0$N_0$ . We call M0$M_0$ by the Franks–Williams manifold. In this paper, we prove that, up to orbit‐equivalence, this DA expanding attractor is the unique expanding attractor supported by N0$N_0$ . We also show that, up to orbit‐equivalence, the non‐transitive Anosov flow constructed by Franks and Williams is the unique non‐transitive Anosov flow supported by M0$M_0$ . We also extend these results to a more general context.
{"title":"Classifying the expanding attractors on the figure‐eight knot exterior and the non‐transitive Anosov flows on the Franks–Williams manifold","authors":"Jiagang Yang, B. Yu","doi":"10.1112/plms.12444","DOIUrl":"https://doi.org/10.1112/plms.12444","url":null,"abstract":"The figure‐eight knot exterior N0$N_0$ supports a natural DA (derived from Anosov) expanding attractor, with which Franks–Williams constructed the first example of non‐transitive Anosov flow. This flow lies in a 3‐manifold M0$M_0$ which is the double of N0$N_0$ . We call M0$M_0$ by the Franks–Williams manifold. In this paper, we prove that, up to orbit‐equivalence, this DA expanding attractor is the unique expanding attractor supported by N0$N_0$ . We also show that, up to orbit‐equivalence, the non‐transitive Anosov flow constructed by Franks and Williams is the unique non‐transitive Anosov flow supported by M0$M_0$ . We also extend these results to a more general context.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45363936","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}
We define an integral form of the deformed W$W$ ‐algebra of type glr${mathfrak {gl}}_r$ , and construct its action on the K$K$ ‐theory groups of moduli spaces of rank r$r$ stable sheaves on a smooth projective surface S$S$ , under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W$W$ ‐algebras by generators and relations is a new feature. Physically, this action encodes the Alday–Gaiotto–Tachikawa correspondence for 5‐dimensional supersymmetric gauge theory on S×$S times$ circle.
{"title":"W$W$ ‐algebras associated to surfaces","authors":"Andrei Neguț","doi":"10.1112/plms.12435","DOIUrl":"https://doi.org/10.1112/plms.12435","url":null,"abstract":"We define an integral form of the deformed W$W$ ‐algebra of type glr${mathfrak {gl}}_r$ , and construct its action on the K$K$ ‐theory groups of moduli spaces of rank r$r$ stable sheaves on a smooth projective surface S$S$ , under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W$W$ ‐algebras by generators and relations is a new feature. Physically, this action encodes the Alday–Gaiotto–Tachikawa correspondence for 5‐dimensional supersymmetric gauge theory on S×$S times$ circle.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47091211","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12410","DOIUrl":"https://doi.org/10.1112/plms.12410","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48667468","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}
M. Bhargava, J. Cremona, T. Fisher, Stevan Gajović
We determine the probability that a random polynomial of degree n$n$ over Zp${mathbb {Z}}_p$ has exactly r$r$ roots in Qp${mathbb {Q}}_p$ , and show that it is given by a rational function of p$p$ that is invariant under replacing p$p$ by 1/p$1/p$ .
{"title":"The density of polynomials of degree n$n$ over Zp${mathbb {Z}}_p$ having exactly r$r$ roots in Qp${mathbb {Q}}_p$","authors":"M. Bhargava, J. Cremona, T. Fisher, Stevan Gajović","doi":"10.1112/plms.12438","DOIUrl":"https://doi.org/10.1112/plms.12438","url":null,"abstract":"We determine the probability that a random polynomial of degree n$n$ over Zp${mathbb {Z}}_p$ has exactly r$r$ roots in Qp${mathbb {Q}}_p$ , and show that it is given by a rational function of p$p$ that is invariant under replacing p$p$ by 1/p$1/p$ .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44819731","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}
We prove the first general finite jet determination result in positive codimension for CR maps from real‐analytic minimal submanifolds M⊂CN$Msubset mathbb {C}^N$ into Nash (real) submanifolds M′⊂CN′$M^{prime }subset mathbb {C}^{N^{prime }}$ . For a sheaf S$mathcal {S}$ of C∞$mathcal {C}^infty$ ‐smooth CR maps from M$M$ into M′$M^{prime }$ , we show that the non‐existence of so‐called 2‐approximate CR S$mathcal {S}$‐deformations from M$M$ into M′$M^{prime }$ implies the following strong finite jet determination property: There exists a map ℓ:M→Z+$ell colon Mrightarrow {mathbb {Z}}_+$ , bounded on compact subsets of M$M$ , such that for every point p∈M$pin M$ , whenever f,g$f,g$ are two elements of Sp$mathcal {S}_p$ with jpℓ(p)f=jpℓ(p)g$j^{ell (p)}_pf=j^{ell (p)}_pg$ , then f=g$f=g$ . Applying the deformation point of view allows a unified treatment of a number of classes of target manifolds, which includes, among others, strictly pseudoconvex, Levi–non‐degenerate, but also some particularly important Levi‐degenerate targets, such as boundaries of classical domains.
{"title":"The finite jet determination problem for CR maps of positive codimension into Nash manifolds","authors":"B. Lamel, N. Mir","doi":"10.1112/plms.12439","DOIUrl":"https://doi.org/10.1112/plms.12439","url":null,"abstract":"We prove the first general finite jet determination result in positive codimension for CR maps from real‐analytic minimal submanifolds M⊂CN$Msubset mathbb {C}^N$ into Nash (real) submanifolds M′⊂CN′$M^{prime }subset mathbb {C}^{N^{prime }}$ . For a sheaf S$mathcal {S}$ of C∞$mathcal {C}^infty$ ‐smooth CR maps from M$M$ into M′$M^{prime }$ , we show that the non‐existence of so‐called 2‐approximate CR S$mathcal {S}$‐deformations from M$M$ into M′$M^{prime }$ implies the following strong finite jet determination property: There exists a map ℓ:M→Z+$ell colon Mrightarrow {mathbb {Z}}_+$ , bounded on compact subsets of M$M$ , such that for every point p∈M$pin M$ , whenever f,g$f,g$ are two elements of Sp$mathcal {S}_p$ with jpℓ(p)f=jpℓ(p)g$j^{ell (p)}_pf=j^{ell (p)}_pg$ , then f=g$f=g$ . Applying the deformation point of view allows a unified treatment of a number of classes of target manifolds, which includes, among others, strictly pseudoconvex, Levi–non‐degenerate, but also some particularly important Levi‐degenerate targets, such as boundaries of classical domains.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44832784","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}
We study the regularity of the viscosity solution u$u$ of the σk$sigma _k$ ‐Loewner–Nirenberg problem on a bounded smooth domain Ω⊂Rn$Omega subset mathbb {R}^n$ for k⩾2$k geqslant 2$ . It was known that u$u$ is locally Lipschitz in Ω$Omega$ . We prove that, with d$d$ being the distance function to ∂Ω$partial Omega$ and δ>0$delta > 0$ sufficiently small, u$u$ is smooth in {0
我们研究了有界光滑域上σk$sigma _k$‐Loewner-Nirenberg问题的黏性解u$u$的规律性Ω∧Rn$Omega 子集mathbb {R}^n$ for k geqslant 2$。已知u$u$是Ω$Omega$中的局部利普希茨函数。我们证明，当d$d$是到∂Ω$partial Omega$和δ> $的距离函数足够小时，u$u$在{0中是光滑的
{"title":"Regularity of viscosity solutions of the σk$sigma _k$ ‐Loewner–Nirenberg problem","authors":"Yanyan Li, Luc Nguyen, Jingang Xiong","doi":"10.1112/plms.12536","DOIUrl":"https://doi.org/10.1112/plms.12536","url":null,"abstract":"We study the regularity of the viscosity solution u$u$ of the σk$sigma _k$ ‐Loewner–Nirenberg problem on a bounded smooth domain Ω⊂Rn$Omega subset mathbb {R}^n$ for k⩾2$k geqslant 2$ . It was known that u$u$ is locally Lipschitz in Ω$Omega$ . We prove that, with d$d$ being the distance function to ∂Ω$partial Omega$ and δ>0$delta > 0$ sufficiently small, u$u$ is smooth in {0","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42365487","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}
For a hypergraph H$H$ , define its intersection spectrum I(H)$I(H)$ as the set of all intersection sizes |E∩F|$|Ecap F|$ of distinct edges E,F∈E(H)$E,Fin E(H)$ . In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a k$k$ ‐uniform 3‐chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with k$k$ . Despite the problem being reiterated several times over the years by Erdős and other researchers, the lower bound of three intersection sizes has remarkably withstood any improvement until now. In this paper, we prove the Erdős–Lovász conjecture in a strong form by showing that there are at least k1/2−o(1)$k^{1/2-o(1)}$ intersection sizes. Our proof consists of a delicate interplay between Ramsey‐type arguments and a density increment approach.
{"title":"The intersection spectrum of 3‐chromatic intersecting hypergraphs","authors":"Matija Bucić, Stefan Glock, B. Sudakov","doi":"10.1112/plms.12436","DOIUrl":"https://doi.org/10.1112/plms.12436","url":null,"abstract":"For a hypergraph H$H$ , define its intersection spectrum I(H)$I(H)$ as the set of all intersection sizes |E∩F|$|Ecap F|$ of distinct edges E,F∈E(H)$E,Fin E(H)$ . In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a k$k$ ‐uniform 3‐chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with k$k$ . Despite the problem being reiterated several times over the years by Erdős and other researchers, the lower bound of three intersection sizes has remarkably withstood any improvement until now. In this paper, we prove the Erdős–Lovász conjecture in a strong form by showing that there are at least k1/2−o(1)$k^{1/2-o(1)}$ intersection sizes. Our proof consists of a delicate interplay between Ramsey‐type arguments and a density increment approach.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46901137","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}
We determine all permutation polynomials over Fq2$mathbb {F}_{q^2}$ of the form XrA(Xq−1)$X^r A(X^{q-1})$ where, for some Q$Q$ that is a power of the characteristic of Fq$mathbb {F}_q$ , we have r≡Q+1(modq+1)$requiv Q+1pmod {q+1}$ and all terms of A(X)$A(X)$ have degrees in {0,1,Q,Q+1}$lbrace 0,1,Q,Q+1rbrace$ . We use this classification to resolve eight conjectures and open problems from the literature, and we list 77 recent results from the literature that follow immediately from the simplest special cases of our result. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high‐degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.
{"title":"Determination of a class of permutation quadrinomials","authors":"Zhiguo Ding, Michael E. Zieve","doi":"10.1112/plms.12540","DOIUrl":"https://doi.org/10.1112/plms.12540","url":null,"abstract":"We determine all permutation polynomials over Fq2$mathbb {F}_{q^2}$ of the form XrA(Xq−1)$X^r A(X^{q-1})$ where, for some Q$Q$ that is a power of the characteristic of Fq$mathbb {F}_q$ , we have r≡Q+1(modq+1)$requiv Q+1pmod {q+1}$ and all terms of A(X)$A(X)$ have degrees in {0,1,Q,Q+1}$lbrace 0,1,Q,Q+1rbrace$ . We use this classification to resolve eight conjectures and open problems from the literature, and we list 77 recent results from the literature that follow immediately from the simplest special cases of our result. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high‐degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49088026","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}
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