{"title":"Vanishing of Brauer classes on K3 surfaces under reduction","authors":"Davesh Maulik, Salim Tayou","doi":"10.1112/jlms.70051","DOIUrl":"https://doi.org/10.1112/jlms.70051","url":null,"abstract":"<p>Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many reductions where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei–Hassett–Várilly-Alvarado.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On some matrix counting problems","authors":"Ali Mohammadi, Alina Ostafe, Igor E. Shparlinski","doi":"10.1112/jlms.70044","DOIUrl":"https://doi.org/10.1112/jlms.70044","url":null,"abstract":"<p>We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>. In particular, in the integer case, we improve a recent bound of V. Blomer and J. Li (2022).</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Witt group of the punctured spectrum of a regular semilocal ring","authors":"Stefan Gille, Ivan Panin","doi":"10.1112/jlms.70042","DOIUrl":"https://doi.org/10.1112/jlms.70042","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> be a regular semilocal ring of dimension <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 <mi>q</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>⩾</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <annotation>$4q+1geqslant 5$</annotation>\u0000 </semantics></math> which contains <span></span><math>\u0000 <semantics>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 <annotation>$frac{1}{2}$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>l</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$lgeqslant 1$</annotation>\u0000 </semantics></math> the number of maximal ideals of <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> which are assumed to be all of the same height, and <span></span><math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$U$</annotation>\u0000 </semantics></math> the punctured spectrum of <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>, that is, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Spec</mo>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 <annotation>$operatorname{Spec}R$</annotation>\u0000 </semantics></math> without the maximal ideals. We show that the Witt ring <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>W</mi>\u0000 <mo>(</mo>\u0000 <mi>U</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{W}(U)$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$U$</annotation>\u0000 </semantics></math> has <span></span><math>\u0000 <semantics>\u0000 <mi>l</mi>\u0000 <annotation>$l$</annotation>\u0000 </semantics></math> non-trivial generators <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>E</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marina Anagnostopoulou-Merkouri, Timothy C. Burness
{"title":"On the regularity number of a finite group and other base-related invariants","authors":"Marina Anagnostopoulou-Merkouri, Timothy C. Burness","doi":"10.1112/jlms.70035","DOIUrl":"https://doi.org/10.1112/jlms.70035","url":null,"abstract":"<p>A <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-tuple <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(H_1, ldots, H_k)$</annotation>\u0000 </semantics></math> of core-free subgroups of a finite group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is said to be regular if <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> has a regular orbit on the Cartesian product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>/</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>×</mo>\u0000 <mi>⋯</mi>\u0000 <mo>×</mo>\u0000 <mi>G</mi>\u0000 <mo>/</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$G/H_1 times cdots times G/H_k$</annotation>\u0000 </semantics></math>. The regularity number of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, denoted by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$R(G)$</annotation>\u0000 </semantics></math>, is the smallest positive integer <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> with the property that every such <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple gro","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marcelo Campos, Gabriel Dahia, João Pedro Marciano
{"title":"On the independence number of sparser random Cayley graphs","authors":"Marcelo Campos, Gabriel Dahia, João Pedro Marciano","doi":"10.1112/jlms.70041","DOIUrl":"https://doi.org/10.1112/jlms.70041","url":null,"abstract":"<p>The Cayley sum graph <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Γ</mi>\u0000 <mi>A</mi>\u0000 </msub>\u0000 <annotation>$Gamma _A$</annotation>\u0000 </semantics></math> of a set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>⊆</mo>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$A subseteq mathbb {Z}_n$</annotation>\u0000 </semantics></math> is defined to have vertex set <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Z}_n$</annotation>\u0000 </semantics></math> and an edge between two distinct vertices <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>∈</mo>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$x, y in mathbb {Z}_n$</annotation>\u0000 </semantics></math> if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>+</mo>\u0000 <mi>y</mi>\u0000 <mo>∈</mo>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 <annotation>$x + y in A$</annotation>\u0000 </semantics></math>. Green and Morris proved that if the set <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> is a <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-random subset of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Z}_n$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p = 1/2$</annotation>\u0000 </semantics></math>, then the independence number of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Γ</mi>\u0000 <mi>A</mi>\u0000 </msub>\u0000 <annotation>$Gamma _A$</annotation>\u0000 </semantics></math> is asymptotically equal to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>(</mo>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142764082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the smooth Whitney fibering conjecture","authors":"C. Murolo, A. du Plessis, D. J. A. Trotman","doi":"10.1112/jlms.70021","DOIUrl":"https://doi.org/10.1112/jlms.70021","url":null,"abstract":"<p>We improve upon the first Thom–Mather isotopy theorem for Whitney stratified sets. In particular, for the more general Bekka stratified sets we show that there is a local foliated structure with continuously varying tangent spaces, thus proving the smooth version of the Whitney fibering conjecture. A regular wing structure is also shown to exist locally, for Bekka stratifications. The proofs involve integrating carefully chosen controlled distributions of vector fields. As an application of our main theorem, we show the density of the subset of strongly topologically stable mappings in the space of all smooth quasi-proper mappings between smooth manifolds, an improvement of a theorem of Mather.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142764081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On dynamical parameter space of cubic polynomials with a parabolic fixed point","authors":"Runze Zhang","doi":"10.1112/jlms.70038","DOIUrl":"https://doi.org/10.1112/jlms.70038","url":null,"abstract":"<p>This article focuses on the connectedness locus of the cubic polynomial slice <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Per</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>λ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathrm{Per}_1(lambda)$</annotation>\u0000 </semantics></math> with a parabolic fixed point of multiplier <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>e</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>π</mi>\u0000 <mi>i</mi>\u0000 <mi>p</mi>\u0000 <mo>/</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$lambda =e^{2pi i{p}/{q}}$</annotation>\u0000 </semantics></math>. We first show that any parabolic component, which is a parallel notion of hyperbolic component, is a Jordan domain. Moreover, a continuum <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 <annotation>$mathcal {K}_lambda$</annotation>\u0000 </semantics></math> called the central part in the connectedness locus is introduced. This is the natural analogue to the closure of the main hyperbolic component of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Per</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathrm{Per}_1(0)$</annotation>\u0000 </semantics></math>. We prove that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 <annotation>$mathcal {K}_lambda$</annotation>\u0000 </semantics></math> is almost a double covering of the filled-in Julia set of the quadratic polynomial <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>λ</mi>\u0000 <mi>z</mi>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>z</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The number of positive solutions for \u0000 \u0000 n\u0000 $n$\u0000 -coupled elliptic systems","authors":"Yongtao Jing, Haidong Liu, Yanyan Liu, Zhaoli Liu, Juncheng Wei","doi":"10.1112/jlms.70040","DOIUrl":"https://doi.org/10.1112/jlms.70040","url":null,"abstract":"<p>We study the number of positive solutions to the <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-coupled elliptic system\u0000\u0000 </p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142762074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}