Michael Allen , Brian Grove , Ling Long, Fang-Ting Tu
{"title":"The explicit hypergeometric-modularity method I","authors":"Michael Allen , Brian Grove , Ling Long, Fang-Ting Tu","doi":"10.1016/j.aim.2025.110411","DOIUrl":"10.1016/j.aim.2025.110411","url":null,"abstract":"<div><div>The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character sums are often congruent or equal to Fourier coefficients of modular forms. In this series of papers, we develop and explore an explicit “Hypergeometric-Modularity” method for associating a modular form to a given hypergeometric datum. In particular, for certain length three and four hypergeometric data we give an explicit method for finding a modular form <em>f</em> such that the corresponding hypergeometric Galois representation has a subrepresentation isomorphic to the Deligne representation of <em>f</em>. Our method utilizes Ramanujan's theory of elliptic functions to alternative bases, commutative formal group laws, and supercongruences. As a byproduct, we give a collection of eta quotients with multiplicative coefficients constructed from hypergeometric functions. In the second paper, we discuss a number of applications, including explicit connections between hypergeometric values and periods of these explicit eta quotients as well as evaluation formulae for certain special <em>L</em>-values.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110411"},"PeriodicalIF":1.5,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322974","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}
William Alexandre , Clifford Gilmore , Sophie Grivaux
{"title":"Typicality of operators on Fréchet algebras admitting a hypercyclic algebra","authors":"William Alexandre , Clifford Gilmore , Sophie Grivaux","doi":"10.1016/j.aim.2025.110406","DOIUrl":"10.1016/j.aim.2025.110406","url":null,"abstract":"<div><div>This paper is devoted to the study of typical properties (in the Baire Category sense) of certain classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence. Our main results show that within natural Polish spaces of continuous operators acting on the algebra <span><math><mi>H</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> of entire functions on <span><math><mi>C</mi></math></span>, a typical operator supports a hypercyclic algebra. We also investigate the case of the complex Fréchet algebras <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>+</mo><mo>∞</mo></math></span>, or <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> endowed with the coordinatewise product, and show that whenever <span><math><mi>M</mi><mo>></mo><mn>1</mn></math></span>, a typical operator on <em>X</em> of norm less than or equal to <em>M</em> admits a hypercyclic algebra.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110406"},"PeriodicalIF":1.5,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306863","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":"CMC hypersurface with finite index in hyperbolic space H4","authors":"Han Hong","doi":"10.1016/j.aim.2025.110408","DOIUrl":"10.1016/j.aim.2025.110408","url":null,"abstract":"<div><div>In this paper, we prove that there are no complete noncompact constant mean curvature hypersurfaces with the mean curvature <span><math><mi>H</mi><mo>></mo><mn>1</mn></math></span>, finite index and finite topology in hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>. A more general nonexistence result can be proved in a 4-dimensional Riemannian manifold with certain curvature conditions. We also show that 4-manifold with <span><math><mi>Ric</mi><mo>></mo><mn>1</mn></math></span> does not contain any complete noncompact minimal stable hypersurface with finite topology.</div><div>The proof relies on the <em>μ</em>-bubble initially introduced by Gromov and further developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110408"},"PeriodicalIF":1.5,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306862","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":"Sharp estimates for the Cramér transform of log-concave measures and geometric applications","authors":"Silouanos Brazitikos , Giorgos Chasapis","doi":"10.1016/j.aim.2025.110407","DOIUrl":"10.1016/j.aim.2025.110407","url":null,"abstract":"<div><div>We establish a new comparison between the Legendre transform of the cumulant generating function and the half-space depth of an arbitrary log-concave probability distribution on the real line, that carries on to the multidimensional setting. Combined with sharp estimates for the Cramér transform of rotationally invariant measures, we are led to some new phase-transition type results for the asymptotics of the expected measure of random polytopes. As a byproduct of our analysis, we address a question on the sharp exponential separability constant for log-concave distributions, in the symmetric case.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110407"},"PeriodicalIF":1.5,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313438","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":"Selfless C*-algebras","authors":"Leonel Robert","doi":"10.1016/j.aim.2025.110409","DOIUrl":"10.1016/j.aim.2025.110409","url":null,"abstract":"<div><div>The aim of this note is to advertise a class of simple C*-algebras which includes noteworthy examples such as the Jiang-Su C*-algebra, the infinite dimensional UHF C*-algebras, the reduced group C*-algebra of the free group in infinitely many generators, and the Cuntz algebras.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110409"},"PeriodicalIF":1.5,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313439","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}
Arkady Berenstein , Azat Gainutdinov , Vassily Gorbounov
{"title":"Generalized electrical Lie algebras","authors":"Arkady Berenstein , Azat Gainutdinov , Vassily Gorbounov","doi":"10.1016/j.aim.2025.110405","DOIUrl":"10.1016/j.aim.2025.110405","url":null,"abstract":"<div><div>We generalize the electrical Lie algebras originally introduced by Lam and Pylyavskyy in several ways. To each Kac-Moody Lie algebra <span><math><mi>g</mi></math></span> we associate two types (vertex type and edge type) of the generalized electrical algebras. The electrical Lie algebras of vertex type are always subalgebras of <span><math><mi>g</mi></math></span> and are flat deformations of the nilpotent Lie subalgebra of <span><math><mi>g</mi></math></span>. In many cases including <span><math><mi>s</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>s</mi><msub><mrow><mi>o</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and <span><math><mi>s</mi><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> we find new (edge) models for our generalized electrical Lie algebras of vertex type. Finding an edge model in general is an interesting open problem.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110405"},"PeriodicalIF":1.5,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144297931","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":"The classification of endotrivial complexes","authors":"Sam K. Miller","doi":"10.1016/j.aim.2025.110404","DOIUrl":"10.1016/j.aim.2025.110404","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group and <em>k</em> a field of prime characteristic <em>p</em>. We give a complete classification of endotrivial complexes, i.e. determine the Picard group <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of the tensor-triangulated category <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>(</mo><mmultiscripts><mrow><mi>triv</mi></mrow><mprescripts></mprescripts><mrow><mi>k</mi><mi>G</mi></mrow><none></none></mmultiscripts><mo>)</mo></math></span>, the bounded homotopy category of <em>p</em>-permutation modules, which Balmer and Gallauer recently considered in <span><span>[8]</span></span>. For <em>p</em>-groups, we identify <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mo>−</mo><mo>)</mo></math></span> with the rational <em>p</em>-biset functor <span><math><mi>C</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mo>−</mo><mo>)</mo></math></span> of Borel-Smith functions and recover a short exact sequence of rational <em>p</em>-biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every <em>p</em>-permutation autoequivalence of a <em>p</em>-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin in <span><span>[22]</span></span>, showing the kernel of the Bouc homomorphism for an arbitrary finite group <em>G</em> is described by superclass functions <span><math><mi>f</mi><mo>:</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>Z</mi></math></span> satisfying the oriented Artin-Borel-Smith conditions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110404"},"PeriodicalIF":1.5,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263041","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":"The space of C1+ac actions of Zd on a one-dimensional manifold is path-connected","authors":"Hélène Eynard-Bontemps , Andrés Navas","doi":"10.1016/j.aim.2025.110395","DOIUrl":"10.1016/j.aim.2025.110395","url":null,"abstract":"<div><div>We show path-connectedness for the space of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> actions by <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> diffeomorphisms with absolutely continuous derivative on both the closed interval and the circle. We also give a new and short proof of the connectedness of the space of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> actions by <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> diffeomorphisms on the interval, as well as an analogous result in the real-analytic setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110395"},"PeriodicalIF":1.5,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255480","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":"On the formula for characteristic determinants of boundary value problems for n × n Dirac type systems and its applications","authors":"Anton A. Lunyov , Mark M. Malamud","doi":"10.1016/j.aim.2025.110389","DOIUrl":"10.1016/j.aim.2025.110389","url":null,"abstract":"<div><div>The paper is concerned with the spectral properties of the boundary value problems (BVP) associated with the following <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Dirac type equation:<span><span><span><math><mo>−</mo><mi>i</mi><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>−</mo><mi>i</mi><mi>Q</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>y</mi><mo>=</mo><mi>λ</mi><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>y</mi><mo>,</mo><mspace></mspace><mi>y</mi><mo>=</mo><mrow><mi>col</mi></mrow><mo>(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>ℓ</mi><mo>]</mo><mo>,</mo></math></span></span></span> on a finite interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>ℓ</mi><mo>]</mo></math></span> subject to the general two-point boundary conditions <span><math><mi>C</mi><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>D</mi><mi>y</mi><mo>(</mo><mi>ℓ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> with <span><math><mi>C</mi><mo>,</mo><mi>D</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>. Here <span><math><mi>Q</mi><mo>=</mo><msubsup><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>j</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is an integrable potential matrix and <span><math><mi>B</mi><mo>=</mo><mrow><mi>diag</mi></mrow><mo>(</mo><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is a diagonal integrable matrix “weight”. If <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></math></span> and <span><math><mi>B</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>=</mo><mrow><mi>diag</mi></mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, this equation turns into <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Dirac equation.</div><div>First, assuming that <span><math><mrow><mi>supp</mi></mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo><mo>⊂</mo><mrow><mi>supp</mi></mrow><mo>(</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span>, we show that the deviation <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>λ</mi><mo>)</mo></math>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110389"},"PeriodicalIF":1.5,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240710","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}
Antonio Avilés , Gonzalo Martínez-Cervantes , Alejandro Poveda , Luis Sáenz
{"title":"A Banach space with L-orthogonal sequences but without L-orthogonal elements","authors":"Antonio Avilés , Gonzalo Martínez-Cervantes , Alejandro Poveda , Luis Sáenz","doi":"10.1016/j.aim.2025.110391","DOIUrl":"10.1016/j.aim.2025.110391","url":null,"abstract":"<div><div>We prove that the existence of Banach spaces with <em>L</em>-orthogonal sequences but without <em>L</em>-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to <span><span>[1, Question 1.1]</span></span>. Generalizing classical <em>Q</em>-point ultrafilters, we introduce the notion of <em>Q</em>-measures and provide several results generalizing former theorems by Miller <span><span>[22]</span></span> and Bartoszynski <span><span>[2]</span></span> for <em>Q</em>-point ultrafilters.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110391"},"PeriodicalIF":1.5,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240707","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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