{"title":"Isotropic constants and regular polytopes","authors":"Christian Kipp","doi":"10.1016/j.aim.2025.110361","DOIUrl":"10.1016/j.aim.2025.110361","url":null,"abstract":"<div><div>We discuss first-order optimality conditions for the isotropic constant and combine them with RS-movements to obtain structural information about polytopal maximizers. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer with a simplicial vertex must be a simplex. A similar statement is shown for a centrally symmetric local maximizer with a simplicial vertex: it has to be a cross-polytope. Moreover, we show that a zonotope that maximizes the isotropic constant and that has a cubical zone must be a cube. Finally, we consider the class of zonotopes with at most <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> generators and determine the extremals in this class.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110361"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115051","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}
Adrian Diaconu , Bogdan Ion , Vicenţiu Paşol , Alexandru A. Popa
{"title":"Residues of quadratic Weyl group multiple Dirichlet series","authors":"Adrian Diaconu , Bogdan Ion , Vicenţiu Paşol , Alexandru A. Popa","doi":"10.1016/j.aim.2025.110359","DOIUrl":"10.1016/j.aim.2025.110359","url":null,"abstract":"<div><div>We give explicit formulas for the residue of the Chinta-Gunnells average attached to a finite irreducible root system, at the polar divisor corresponding to a simple short root. The formula describes the residue in terms of the average attached to the root subsystem orthogonal to the relevant simple root. As a consequence, we obtain similar formulas for the residues of quadratic Weyl group multiple Dirichlet series over the rational function field and over the Gaussian field. The residue formula also allows us to obtain a new expression for the Chinta-Gunnells average of a finite irreducible root system, as an average over a maximal parabolic subgroup of a rational function that has an explicit description reflecting the combinatorics of the root system.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110359"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115052","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":"Cohomology of (φ,τ)-modules","authors":"Hui Gao , Luming Zhao","doi":"10.1016/j.aim.2025.110363","DOIUrl":"10.1016/j.aim.2025.110363","url":null,"abstract":"<div><div>We construct cohomology theories for <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>-modules, and study their relation with cohomology of <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>Γ</mi><mo>)</mo></math></span>-modules, as well as Galois cohomology. The method is axiomatic, and can treat the étale case, the overconvergent case, and the rigid-overconvergent case simultaneously. We use recent advances in locally analytic cohomology as a key ingredient.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110363"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115053","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":"Towards a generic absoluteness theorem for Chang models","authors":"Sandra Müller , Grigor Sargsyan","doi":"10.1016/j.aim.2025.110357","DOIUrl":"10.1016/j.aim.2025.110357","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> be the set of all universally Baire sets of reals. Inspired by the work done in <span><span>[22]</span></span> and <span><span>[21]</span></span>, we introduce a new technique for establishing generic absoluteness results for models containing <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>.</div><div>Our main technical tool is an iteration that realizes <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> as the sets of reals in a derived model of some iterate of <em>V</em>. We show, from a supercompact cardinal <em>κ</em> and a proper class of Woodin cardinals, that whenever <span><math><mi>g</mi><mo>⊆</mo><mi>Col</mi><mo>(</mo><mi>ω</mi><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>κ</mi></mrow></msup></mrow></msup><mo>)</mo></math></span> is <em>V</em>-generic and <em>h</em> is <span><math><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo></math></span>-generic for some poset <span><math><mi>P</mi><mo>∈</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo></math></span>, there is an elementary embedding <span><math><mi>j</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>M</mi></math></span> such that <span><math><mi>j</mi><mo>(</mo><mi>κ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>V</mi><mo>[</mo><mi>g</mi><mo>⁎</mo><mi>h</mi><mo>]</mo></mrow></msubsup></math></span> and <span><math><mi>L</mi><mo>(</mo><msup><mrow><mi>Γ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>,</mo><mi>R</mi><mo>)</mo></math></span> as computed in <span><math><mi>V</mi><mo>[</mo><mi>g</mi><mo>⁎</mo><mi>h</mi><mo>]</mo></math></span> is a derived model of <em>M</em> at <span><math><mi>j</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span>. Here <em>j</em> is obtained by iteratively taking ultrapowers of <em>V</em> by extenders with critical point <em>κ</em> and its images.</div><div>As a corollary we obtain that <span><math><mi>Sealing</mi></math></span> holds in <span><math><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo></math></span>, which was previously demonstrated by Woodin using the stationary tower forcing. Also, using a theorem of Woodin, we conclude that the derived model of <em>V</em> at <em>κ</em> satisfies <span><math><msub><mrow><mi>AD</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>+</mo><mtext>“</mtext><mi>Θ</mi></math></span> is a regular cardinal”.</div><div>Inspired by core model induction, we introduce the definable powerset <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> of <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> and use our derived model representation mentioned above to show that the theory of <span><math><mi>L</mi><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo></math></span> cannot be changed by forcing (see <span><span>Theorem 1","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110357"},"PeriodicalIF":1.5,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106163","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":"Five-variable p-adic L-functions for U(3)×U(2)","authors":"Ming-Lun Hsieh , Shunsuke Yamana","doi":"10.1016/j.aim.2025.110355","DOIUrl":"10.1016/j.aim.2025.110355","url":null,"abstract":"<div><div>We construct a five-variable <em>p</em>-adic <em>L</em>-function attached to Hida families on the definite unitary groups <span><math><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span> by using the Ichino-Ikeda formula. The interpolation formula fits into the conjectural shape of <em>p</em>-adic <em>L</em>-functions predicted by Coates and Perrin-Riou.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110355"},"PeriodicalIF":1.5,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099153","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}
Andriy Regeta , Christian Urech , Immanuel van Santen
{"title":"The structure of algebraic families of birational transformations","authors":"Andriy Regeta , Christian Urech , Immanuel van Santen","doi":"10.1016/j.aim.2025.110354","DOIUrl":"10.1016/j.aim.2025.110354","url":null,"abstract":"<div><div>We give a description of the algebraic families of birational transformations of an algebraic variety <em>X</em>. As an application, we show that the morphisms to <span><math><mi>Bir</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of <span><math><mi>Bir</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> are exactly the closed finite-dimensional subgroups with finitely many components. We also study algebraic families of birational transformations preserving a fibration. This builds on previous work of Blanc-Furter <span><span>[2]</span></span>, Hanamura <span><span>[9]</span></span>, and Ramanujam <span><span>[20]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110354"},"PeriodicalIF":1.5,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099154","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":"Linkage principle for small quantum groups","authors":"Cristian Vay","doi":"10.1016/j.aim.2025.110347","DOIUrl":"10.1016/j.aim.2025.110347","url":null,"abstract":"<div><div>We consider small quantum groups with root systems of Cartan, super and modular type, among others. These are constructed as Drinfeld doubles of finite-dimensional Nichols algebras of diagonal type. We prove a linkage principle for them by adapting techniques from the work of Andersen, Jantzen and Soergel in the context of small quantum groups at roots of unity. Consequently we characterize the blocks of the category of modules. We also find a notion of (a)typicality similar to the one in the representation theory of Lie superalgebras. The typical simple modules turn out to be the simple and projective Verma modules. Moreover, we deduce a character formula for 1-atypical simple modules.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110347"},"PeriodicalIF":1.5,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068795","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":"Asymptotic growth of translation-dilation orbits","authors":"Victor Y. Wang","doi":"10.1016/j.aim.2025.110341","DOIUrl":"10.1016/j.aim.2025.110341","url":null,"abstract":"<div><div>By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain elusive in general.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110341"},"PeriodicalIF":1.5,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069244","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":"Helly-type theorems, CAT(0) spaces, and actions of automorphism groups of free groups","authors":"Martin R. Bridson","doi":"10.1016/j.aim.2025.110343","DOIUrl":"10.1016/j.aim.2025.110343","url":null,"abstract":"<div><div>We prove a variety of fixed-point theorems for groups acting on CAT(0) spaces. Fixed points are obtained by a bootstrapping technique, whereby increasingly large subgroups are proved to have fixed points: specific configurations in the subgroup lattice of Γ are exhibited and Helly-type theorems are developed to prove that the fixed-point sets of the subgroups in the configuration intersect. In this way, we obtain lower bounds on the smallest dimension <span><math><mrow><mi>FixDim</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> in which various groups of geometric interest can act on a complete CAT(0) space without a global fixed point. For automorphism groups of free groups, we prove <span><math><mrow><mi>FixDim</mi></mrow><mo>(</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>)</mo><mo>≥</mo><mo>⌊</mo><mn>2</mn><mi>n</mi><mo>/</mo><mn>3</mn><mo>⌋</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110343"},"PeriodicalIF":1.5,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068794","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":"Local well-posedness of the incompressible current-vortex sheet problems","authors":"Sicheng Liu , Zhouping Xin","doi":"10.1016/j.aim.2025.110339","DOIUrl":"10.1016/j.aim.2025.110339","url":null,"abstract":"<div><div>We prove the local well-posedness of the incompressible current-vortex sheet problems in standard Sobolev spaces under the surface tension or the Syrovatskij condition, which shows that both capillary forces and large tangential magnetic fields can stabilize the motion of current-vortex sheets. Furthermore, under the Syrovatskij condition, the vanishing surface tension limit is established for the motion of current-vortex sheets. These results hold without assuming the interface separating the two plasmas being a graph.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110339"},"PeriodicalIF":1.5,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068796","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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