Advances in Mathematics最新文献

{"title":"Period function of Maass forms from Ramanujan's lost notebook","authors":"YoungJu Choie ,&nbsp;Rahul Kumar","doi":"10.1016/j.aim.2025.110317","DOIUrl":"10.1016/j.aim.2025.110317","url":null,"abstract":"<div><div>The Lost Notebook of Ramanujan contains a number of beautiful formulas, one of which can be found on its page 220. It involves an interesting function, which we denote as <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. In this paper, we show that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> belongs to the category of period functions as it satisfies the period relations of Maass forms in the sense of Lewis and Zagier <span><span>[11]</span></span>. Hence, we refer to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> as the <em>Ramanujan period function</em>. Moreover, one of the salient aspects of the Ramanujan period function <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> that we found out is that it is a Hecke eigenfunction under the action of Hecke operators on the space of periods. We also establish that it naturally appears in a Kronecker limit formula of a certain zeta function, revealing its connections to various topics. Finally, we generalize <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> to include a parameter <em>s</em>, connecting our work to the broader theory of period functions developed by Bettin and Conrey <span><span>[4]</span></span> and Lewis and Zagier <span><span>[11]</span></span>. We emphasize that Ramanujan was the first to study this function, marking the beginning of the study of period functions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110317"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907567","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":"A dimensional mass transference principle for Borel probability measures and applications","authors":"Edouard Daviaud","doi":"10.1016/j.aim.2025.110304","DOIUrl":"10.1016/j.aim.2025.110304","url":null,"abstract":"<div><div>In this article, we establish a dimensional mass transference principle valid when the ambient measure is finite. We provide two applications of this result. First we study certain dynamical coverings associated with some self-similar IFS with overlaps and then we give an application in Diophantine approximation to rational approximation among points of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> with few digit equal to 1 in their base-3 expansion.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110304"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907564","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":"Lower bounds on Loewy lengths of modules of finite projective dimension","authors":"Nawaj KC ,&nbsp;Josh Pollitz","doi":"10.1016/j.aim.2025.110309","DOIUrl":"10.1016/j.aim.2025.110309","url":null,"abstract":"<div><div>This article is concerned with nonzero modules of finite length and finite projective dimension over a local ring. We show the Loewy length of such a module is larger than the regularity of the ring whenever the ring is strict Cohen-Macaulay, establishing a conjecture of Corso–Huneke–Polini–Ulrich for such rings. In fact, we show the stronger result that the Loewy length of a nonzero module of finite flat dimension is at least the regularity for strict Cohen-Macaulay rings, which significantly strengthens a theorem of Avramov–Buchweitz–Iyengar–Miller. As an application we simultaneously verify a Lech-like conjecture, comparing generalized Loewy length along flat local extensions, and a conjecture of Hanes for strict Cohen-Macaulay rings. Finally, we also give notable improvements to known lower bounds for Loewy lengths without the strict Cohen-Macaulay assumption. The strongest general bounds we achieve are over complete intersection rings.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110309"},"PeriodicalIF":1.5,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143896107","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":"Logarithmic tautological rings of the moduli spaces of curves","authors":"Rahul Pandharipande,&nbsp;Dhruv Ranganathan,&nbsp;Johannes Schmitt,&nbsp;Pim Spelier","doi":"10.1016/j.aim.2025.110291","DOIUrl":"10.1016/j.aim.2025.110291","url":null,"abstract":"<div><div>We define the logarithmic tautological rings of the moduli spaces of Deligne–Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras are infinite dimensional, a connection to polyhedral combinatorics via a new theory of homological piecewise polynomials allows an effective study. A complete calculation is given in genus 0 via the algebra of piecewise polynomials on the cone stack of the associated Artin fan (lifting Keel's presentation of the Chow ring of <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msub></math></span>). Counterexamples to the simplest generalizations in genus 1 are presented. We show, however, that the structure of the log tautological rings is determined by the complete knowledge of all relations in the standard tautological rings of the moduli spaces of curves. In particular, Pixton's conjecture concerning relations in the standard tautological rings lifts to a complete conjecture for relations in the log tautological rings of the moduli spaces of curves. Several open questions are discussed.</div><div>We develop the entire theory of logarithmic tautological classes in the context of arbitrary smooth normal crossings pairs <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> with explicit formulas for intersection products. As a special case, we give an explicit set of additive generators of the full logarithmic Chow ring of <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> in terms of Chow classes on the strata of <em>X</em> and piecewise polynomials on the cone stack.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110291"},"PeriodicalIF":1.5,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143898860","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":"Strong cohomological rigidity of Bott manifolds","authors":"Suyoung Choi ,&nbsp;Taekgyu Hwang ,&nbsp;Hyeontae Jang","doi":"10.1016/j.aim.2025.110305","DOIUrl":"10.1016/j.aim.2025.110305","url":null,"abstract":"<div><div>We show that two Bott manifolds are diffeomorphic if and only if their integral cohomology rings are isomorphic as graded rings. In fact, we prove that any graded cohomology ring isomorphism between two Bott manifolds is induced by a diffeomorphism.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110305"},"PeriodicalIF":1.5,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882154","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":"Projection algebras and free projection- and idempotent-generated regular ⁎-semigroups","authors":"James East ,&nbsp;Robert D. Gray ,&nbsp;P.A. Azeef Muhammed ,&nbsp;Nik Ruškuc","doi":"10.1016/j.aim.2025.110288","DOIUrl":"10.1016/j.aim.2025.110288","url":null,"abstract":"<div><div>The purpose of this paper is to introduce a new family of semigroups—the free projection-generated regular ⁎-semigroups—and initiate their systematic study. Such a semigroup <span><math><mtext>PG</mtext><mo>(</mo><mi>P</mi><mo>)</mo></math></span> is constructed from a projection algebra <em>P</em>, using the recent groupoid approach to regular ⁎-semigroups. The assignment <span><math><mi>P</mi><mo>↦</mo><mtext>PG</mtext><mo>(</mo><mi>P</mi><mo>)</mo></math></span> is a left adjoint to the forgetful functor that maps a regular ⁎-semigroup <em>S</em> to its projection algebra <span><math><mi>P</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>. In fact, the category of projection algebras is coreflective in the category of regular ⁎-semigroups. The algebra <span><math><mi>P</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span> uniquely determines the biordered structure of the idempotents <span><math><mi>E</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>, up to isomorphism, and this leads to a category equivalence between projection algebras and regular ⁎-biordered sets. As a consequence, <span><math><mtext>PG</mtext><mo>(</mo><mi>P</mi><mo>)</mo></math></span> can be viewed as a quotient of the classical free idempotent-generated (regular) semigroups <span><math><mtext>IG</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span> and <span><math><mtext>RIG</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span>, where <span><math><mi>E</mi><mo>=</mo><mi>E</mi><mo>(</mo><mtext>PG</mtext><mo>(</mo><mi>P</mi><mo>)</mo><mo>)</mo></math></span>; this is witnessed by a number of presentations in terms of generators and defining relations. The semigroup <span><math><mtext>PG</mtext><mo>(</mo><mi>P</mi><mo>)</mo></math></span> can also be interpreted topologically, through a natural link to the fundamental groupoid of a simplicial complex explicitly constructed from <em>P</em>. The above theory is illustrated on a number of examples. In one direction, the free construction applied to the projection algebras of adjacency semigroups yields a new family of graph-based path semigroups. In another, it turns out that, remarkably, the Temperley–Lieb monoid <span><math><mi>T</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the free regular ⁎-semigroup over its own projection algebra <span><math><mi>P</mi><mo>(</mo><mi>T</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110288"},"PeriodicalIF":1.5,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886060","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":"Affine quermassintegrals and even Minkowski valuations","authors":"Georg C. Hofstätter,&nbsp;Philipp Kniefacz,&nbsp;Franz E. Schuster","doi":"10.1016/j.aim.2025.110285","DOIUrl":"10.1016/j.aim.2025.110285","url":null,"abstract":"<div><div>It is shown that each continuous even Minkowski valuation on convex bodies of degree <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> intertwining rigid motions is obtained from convolution of the <em>i</em>th projection function with a unique spherical Crofton distribution. In case of a non-negative distribution, the polar volume of the associated Minkowski valuation gives rise to an isoperimetric inequality which strengthens the classical relation between the <em>i</em>th quermassintegral and the volume. This large family of inequalities unifies earlier results obtained for <span><math><mi>i</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. In these cases, isoperimetric inequalities for affine quermassintegrals, specifically the Blaschke–Santaló inequality for <span><math><mi>i</mi><mo>=</mo><mn>1</mn></math></span> and the Petty projection inequality for <span><math><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, were proven to be the strongest inequalities. An analogous result for the intermediate degrees is established here. Finally, a new sufficient condition for the existence of maximizers for the polar volume of Minkowski valuations intertwining rigid motions reveals unexpected examples of volume inequalities having asymmetric extremizers.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110285"},"PeriodicalIF":1.5,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882163","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":"Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions","authors":"Atul Dixit,&nbsp;Sumukha Sathyanarayana ,&nbsp;N. Guru Sharan","doi":"10.1016/j.aim.2025.110303","DOIUrl":"10.1016/j.aim.2025.110303","url":null,"abstract":"<div><div>The Mordell-Tornheim zeta function and the Herglotz-Zagier function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are two important functions in Mathematics. By generalizing a special case of the former, namely <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>, we show that the theories of these functions are inextricably woven. We obtain a three-term functional equation for <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> as well as decompose it in terms of the Herglotz-Hurwitz function <span><math><mi>Φ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>. This decomposition can be conceived as a two-term functional equation for <span><math><mi>Φ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>. Through this result, we are not only able to get Zagier's identity relating <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> with <span><math><mi>F</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>)</mo></math></span> but also a two-term functional equation for Ishibashi's generalization of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, namely, <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, which has been sought after for over twenty years. We further generalize <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> by incorporating two Gauss sums, each associated to a Dirichlet character, and decompose it in terms of an interesting integral which involves the Fekete polynomial as well as the character polylogarithm. This result gives infinite families of functional equations of Herglotz-type integrals out of which only two, due to Choie and Kumar, were known so far. The first one among the two involves the integral <span><math><mi>J</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> whose special values have received a lot of attention, more recently, in the work of Muzzaffar and Williams, and in that of Radchenko and Zagier. Analytic continuation of our generalization of <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> is also accomplished which allows us to obtain transformations between certain double series and Herglotz-type integrals or their explicit evaluations.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110303"},"PeriodicalIF":1.5,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879396","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 subspaces of indecomposable Banach spaces","authors":"Piotr Koszmider,&nbsp;Zdeněk Silber","doi":"10.1016/j.aim.2025.110292","DOIUrl":"10.1016/j.aim.2025.110292","url":null,"abstract":"<div><div>We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> as a quotient (equivalently do not admit a subspace isomorphic to <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>c</mi><mo>)</mo></math></span>). This includes all Asplund spaces and all weakly Lindelöf determined Banach spaces of density not bigger than the continuum. However, we also show that this class includes some Banach spaces admitting <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> as a quotient. This sheds some light on the question asked in the paper [2] of Argyros and Haydon whether all Banach spaces not containing <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> embed in some indecomposable Banach spaces. Our method of constructing indecomposable Banach spaces above a given Banach space is a considerable modification of the method of constructing Banach spaces of continuous functions with few^⁎ operators developed before by the first-named author.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110292"},"PeriodicalIF":1.5,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879395","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":"Unimodal sequences and mixed false theta functions","authors":"Kevin Allen,&nbsp;Robert Osburn","doi":"10.1016/j.aim.2025.110293","DOIUrl":"10.1016/j.aim.2025.110293","url":null,"abstract":"<div><div>We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions. These results complement recent striking work of Mortenson and Zwegers on the mixed mock modularity of the generalized <em>U</em>-function due to Hikami and Lovejoy. As an application, we demonstrate how to recover classical partial theta function identities which appear in Ramanujan's lost notebook and in work of Warnaar.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110293"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143873941","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}