Annales Mathematiques du Quebec最新文献

{"title":"Paramètres dans les corps algébriquement clos","authors":"Bruno Poizat","doi":"10.1007/s40316-025-00242-1","DOIUrl":"10.1007/s40316-025-00242-1","url":null,"abstract":"<p> Ce papier est une incursion de la Théorie des Modèles dans la Géométrie Algébrique, et son auteur attache un plus grand prix aux méthodes employées pour parvenir aux résultats obtenus qu'à la valeur intrinsèque de ces résultats-mêmes. En témoigne la nature des questions réparties dans le texte. On y étudie les influences réciproques du groupe des automorphismes d'un corps algébriquement clos K sur le groupe des automorphismes d'une structure S définissable dans K. Les paramètres nécessaires aux définitions vont y jouer un rôle de premier plan, ainsi que les propriétés très particulières de la Théorie des Modèles des corps algébriquement clos. Il commence par un commentaire d’un résultat de A.V. Borovik, qui a été la source de son inspiration, mettant en évidence sa dépendance au célèbre Théorème de Borel et Tits sur les isomorphismes abstraits des groupes algébriques simples, considéré d’un point de vue modèle-théorique. Ce théorème conduit finalement à la description des automorphismes d'ordre fini d'un groupe algébrique simple (sur un corps de base algébriquement clos), et de ses groupes superstables d’automorphismes, qui est basée sur des arguments généraux de Théorie des Modèles, ne demandant qu’une inspection minimale de la structure du groupe; pour en tirer des conséquences, nous devons affermir un argument elliptique d’Altinel, Borovik et Cherlin, dans une démonstration qui est pourtant cruciale dans leur contexte inductif. En fait, notre version du Théorème de Borel et Tits ne dépend pas de la présence d’une loi de groupe; il est valable plus généralement pour ce que nous appelons les <i>structures constructives autonomes,</i> qui sont les structures infinies S, définissables dans un corps algébriquement clos K, pour lesquelles tout ce qui est définissable sur S dans le langage du corps K est définissable (avec paramètres) dans le langage de S. Un exemple significatif de telles structures est donné par les <i>multicorps,</i> dont la banale définition cache une subtile théorie de Galois en caractéristique p; en effet, dans n’importe quelle structure constructible autonome S on peut définir sans paramètres un multicorps qui contrôle les automorphismes de S au sens suivant: ceux d’entre eux dont l’action sur ce multicorps est d’ordre fini forment un groupe, noté Aut<sub>max</sub>(S), qui est à la fois le plus grand groupe définissable d’automorphismes de S, et le plus grand groupe superstable d’automorphismes de S. En caractéristique nulle, ce résultat est facile à établir, car on peut alors définir sans paramètres dans S un unicorps, c’est-à-dire une copie L du corps de base K. Quand S est un groupe algébrique simple G, dans les cas ordinaires une telle copie L existe même en caractéristique p; mais il y a des cas spéciaux, dus à la présence d'endogénies exceptionnelles, ou seulement un bicorps (L<sub>1</sub>,L<sub>2</sub>) est définissable sans paramètres; Aut<sub>max</sub>(G) est le noyau de l’action des automorphismes de G sur le cor","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"295 - 321"},"PeriodicalIF":0.4,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The de Rham period map for punctured elliptic curves and the KZB equation","authors":"Ben Moore","doi":"10.1007/s40316-025-00246-x","DOIUrl":"10.1007/s40316-025-00246-x","url":null,"abstract":"<div><p>We demonstrate that the algebraic KZB connection of Levin–Racinet and Luo on a once-punctured elliptic curve represents Kim’s universal unipotent connection, and we observe that the Hodge filtration on the KZB connection has a particularly simple form. This allows us to generalise previous work of Beacom by writing down explicitly the maximal metabelian quotient of Kim’s de Rham period map in terms of elliptic polylogarithms. As far as we are aware this is the first time that the de Rham period map has been written out for an infinite dimensional quotient of the de Rham fundamental group on any curve of positive genus.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"167 - 184"},"PeriodicalIF":0.4,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Iwasawa theory for weighted graphs","authors":"Taiga Adachi,&nbsp;Kosuke Mizuno,&nbsp;Sohei Tateno","doi":"10.1007/s40316-025-00247-w","DOIUrl":"10.1007/s40316-025-00247-w","url":null,"abstract":"<div><p>Let <i>p</i> be a prime number and let <i>d</i> be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa’s class number formula and that of Kida’s formula for compatible systems of <span>((mathbb {Z}/p^nmathbb {Z})^d)</span>-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas of Iwasawa theory to the theory of discrete-time quantum walks in graphs.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"231 - 265"},"PeriodicalIF":0.4,"publicationDate":"2025-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Noncommutative differential geometry on infinitesimal spaces","authors":"Damien Tageddine,&nbsp;Jean-Christophe Nave","doi":"10.1007/s40316-025-00264-9","DOIUrl":"10.1007/s40316-025-00264-9","url":null,"abstract":"<div><p>In this paper, we use noncommutative differential geometry to formalize a framework for discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a <span>(C^*)</span>-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function <i>C</i>(<i>M</i>) over a manifold <i>M</i> can be approximated by a direct limit of <span>(C^*)</span>-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the <i>d</i>-lattice in <span>(mathbb {R}^d)</span> and for the torus <span>(mathbb {T}^d)</span>.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"1 - 33"},"PeriodicalIF":0.4,"publicationDate":"2025-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Poisson structure on character varieties, II","authors":"Indranil Biswas,&nbsp;Lisa C. Jeffrey","doi":"10.1007/s40316-025-00256-9","DOIUrl":"10.1007/s40316-025-00256-9","url":null,"abstract":"<div><p>Let <i>G</i> be a complex reductive group and <span>(D, subset , X)</span> a finite subset of a compact Riemann surface <i>X</i>. It was shown in Biswas and Jeffrey (Ann Math Québec 45: 213–219, 2021) that the moduli space of <i>G</i>–characters of <span>(pi _1(X{setminus } D))</span> has a natural Poisson structure. We show that the moduli space of logarithmic <i>G</i>–connections on <i>X</i> singular over <i>D</i> has a Poisson structure. It is proved that the monodromy map from the moduli space of logarithmic <i>G</i>–connections to the moduli space of <i>G</i>–characters is Poisson structure preserving.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"267 - 276"},"PeriodicalIF":0.4,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symplectically self-polar polytopes of minimal capacity","authors":"Mark Berezovik","doi":"10.1007/s40316-025-00251-0","DOIUrl":"10.1007/s40316-025-00251-0","url":null,"abstract":"<div><p>In this paper we continue the study of symplectically self-polar convex bodies started in [3]. We construct symplectically self-polar convex bodies of the minimal Ekeland–Hofer–Zehnder capacity. This in turn proves that the lower bound for the Ekeland–Hofer–Zehnder capacity for centrally symmetric convex bodies obtained in [1] cannot be improved. We also make some numerical experiments and speculations regarding the minimal volume of symplectically self-polar convex bodies.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"335 - 353"},"PeriodicalIF":0.4,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-025-00251-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The multiplicative formula of langlands for orbital integrals in GL(2)","authors":"Malors Espinosa","doi":"10.1007/s40316-025-00259-6","DOIUrl":"10.1007/s40316-025-00259-6","url":null,"abstract":"<div><p>In Langlands (Contributions to automorphic forms, geometry, and number theory. Johns Hopkins University Press, Baltimore, pp. 611–697, 2004.), Langlands introduces a formula for a specific product of orbital integrals in <span>(text{ GL }(2, mathbb {Q}))</span>. Altuğ (Compos Math 151(10), 1791–1820, 2015), employs this formula to manipulate the regular elliptic part of the trace formula, with the aim of eliminating the contribution of the trivial representation from the spectral side. In (Arthur, Adv Math 327:425–469, 2018), Arthur predicts that this formula coincides with a product of polynomials associated with zeta functions of orders developed by Yun (Ramanujan Math Soc Lect Notes 20:399–420, 2013). In (Espinosa, J Number Theory 252:379–404, 2023), we determined the explicit polynomials for the relevant quadratic orders. This paper demonstrates how these polynomials can effectively generalize Langlands’ formula to <i>GL</i>(2, <i>K</i>), for general algebraic number fields <i>K</i>. Furthermore, we also use this formula to extend a well-known formula of Zagier to any algebraic number field. <b>Résumé.</b> Langlands a trouvé une formule pour un produit d’integrales orbitales sur <span>(text{ GL }(2, mathbb {Q}))</span> [8]. Altuğ a employé cette formule pour gérer la partie régulière elliptique de la formule de trace, avec l’objectif d’annuler la contribution de la représentation triviale de la composante spectrale [1]. Arthur a conjecturé [2] que cette formule est le produit des polynômes associés aux fonctions zêta d’ordres developé par Zhiwei Yun dans [10]. Dans [5], on a calculé ces polynômes pour les ordres quadratiques pertinents. Dans cet article, on prouve une généralization de la formule de Langlands pour GL(2,K), où K est un corps de nombres quelconque, en utilisant les polynômes. De plus, on utilise cette formule pour vérifier la formule de Zagier dans un corps de nombres quelconque.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"201 - 229"},"PeriodicalIF":0.4,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds","authors":"Tirumala Chakradhar","doi":"10.1007/s40316-025-00255-w","DOIUrl":"10.1007/s40316-025-00255-w","url":null,"abstract":"<div><p>We consider the Steklov problem on differential <span>(ptext {-})</span>forms defined by Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., <span>(0text {-})</span>forms), highlighting the influence of the underlying topology on the spectrum for <span>(ptext {-})</span>forms in general.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"421 - 443"},"PeriodicalIF":0.4,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-025-00255-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Functional equations of algebraic Rankin–Selberg p-adic L-functions","authors":"Kâzım Büyükboduk,&nbsp;Manisha Ganguly","doi":"10.1007/s40316-025-00250-1","DOIUrl":"10.1007/s40316-025-00250-1","url":null,"abstract":"<div><p>This article presents an approach to the functional equation for Selmer complexes, which in turn have applications in the Iwasawa theoretic study of Rankin–Selberg products of the Hida and Coleman families. Our treatment establishes the functional equation for algebraic <i>p</i>-adic <i>L</i>-functions (that are given in terms of characteristic ideals of Selmer groups arising as the cohomology of appropriately defined Selmer complexes in degree 2). This is achieved by recovering the characteristic ideal as the determinant of the said Selmer complex, once we prove (under suitable but rather mild hypotheses) that the Selmer complex in question is perfect with amplitude [1, 2], and its cohomology is concentrated in degree-2. The perfectness of these Selmer complexes turns out to be a delicate problem, and the required properties require a study of Tamagawa numbers in families, which may be of independent interest.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"93 - 142"},"PeriodicalIF":0.4,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-025-00250-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polyakov formulas for conical singularities in two dimensions","authors":"Clara L. Aldana,&nbsp;Klaus Kirsten,&nbsp;Julie Rowlett","doi":"10.1007/s40316-025-00263-w","DOIUrl":"10.1007/s40316-025-00263-w","url":null,"abstract":"<div><p>We investigate the zeta-regularized determinant and its variation in the presence of conical singularities, boundaries, and corners. For surfaces with isolated conical singularities which may also have one or more smooth boundary components, we demonstrate both a variational Polyakov formula as well as an integrated Polyakov formula for the conformal variation of the Riemannian metric with conformal factors which are smooth up to all singular points and boundary components. We demonstrate the analogous result for curvilinear polygonal domains in surfaces. We then specialize to finite circular sectors and cones and via two independent methods obtain variational Polyakov formulas for the dependence of the determinant on the opening angle. Notably, this requires the conformal factor to be logarithmically singular at the vertex. We further obtain explicit formulas for the determinant for finite circular sectors and cones.\u0000</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"35 - 72"},"PeriodicalIF":0.4,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-025-00263-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}