{"title":"Explicit count of integral ideals of an imaginary quadratic field","authors":"Olivier Ramaré","doi":"10.1007/s40316-025-00243-0","DOIUrl":"10.1007/s40316-025-00243-0","url":null,"abstract":"<div><p>We provide explicit bounds for the number of integral ideals of norms at most <i>X</i> in <span>(mathbb {Q}[sqrt{d}])</span> when <span>(d <0)</span> is a fundamental discriminant with an error term of size <span>(mathcal {O}(X^{1/3}))</span>. In particular, we prove that, when <span>(chi )</span> is the non-principal character modulo 3 and <span>(Xge 1)</span>, we have <span>(sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{3sqrt{3}} +mathcal {O}^*(1.94,X^{1/3}))</span>, and that, when <span>(chi )</span> is the non-principal character modulo 4 and <span>(Xge 1)</span>, we have <span>(sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{4} +mathcal {O}^*(1.4,X^{1/3}))</span>. <b>Résumé.</b> Nous dénombrons de façon explicite avec un terme d’erreur <span>(mathcal {O}(X^{1/3}))</span> le nombre d’idéaux entiers de norme au plus <i>X</i> du corps <span>(mathbb {Q}[sqrt{d}])</span> lorsque <span>(d <0)</span> est un discriminant fondamental. Nous montrons en particulier que, lorsque <span>(chi )</span> est le caractère non principal modulo 3 et <span>(Xge 1)</span>, nous avons <span>(sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{3sqrt{3}} +mathcal {O}^*(1.94,X^{1/3}))</span>, et que , lorsque <span>(chi )</span> est le caractère non principal modulo 4 et <span>(Xge 1)</span>, nous avons <span>(sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{4} +mathcal {O}^*(1.4,X^{1/3} ))</span>.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"463 - 475"},"PeriodicalIF":0.4,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223712","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":"Contractible complexes and non-positive immersions","authors":"William Fisher","doi":"10.1007/s40316-024-00229-4","DOIUrl":"10.1007/s40316-024-00229-4","url":null,"abstract":"<div><p>We provide examples of contractible complexes which fail to have non-positive immersions and weak non-positive immersions, answering a conjecture of Wise in the negative.\u0000</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"367 - 373"},"PeriodicalIF":0.4,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-024-00229-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223710","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":"Galois codescent for motivic tame kernels","authors":"J. Assim, A. Movahhedi","doi":"10.1007/s40316-024-00233-8","DOIUrl":"10.1007/s40316-024-00233-8","url":null,"abstract":"<div><p>Let <i>L</i>/<i>F</i> be a finite Galois extension of number fields with an arbitrary Galois group <i>G</i>. We give an explicit description of the kernel of the natural map on motivic cohomology of the rings of integers <span>(H^2_mathcal {M}(o_L, {textbf{Z}}(i))_{G} {longrightarrow } H^2_mathcal {M}(o_F, {textbf{Z}}(i)))</span>. Using the link between motivic cohomology and <i>K</i>-theory, we deduce genus formulae for all even <i>K</i>-groups <span>(K_{2i-2}(o_F))</span> of the ring of integers. As a by-product, we answer a question raised by B. Kahn about a signature map.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"503 - 555"},"PeriodicalIF":0.4,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223724","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":"Radial limits of solutions to elliptic partial differential equations","authors":"Paul M. Gauthier, Mohammad Shirazi","doi":"10.1007/s40316-025-00241-2","DOIUrl":"10.1007/s40316-025-00241-2","url":null,"abstract":"<div><p>For certain elliptic differential operators <i>L</i>, we study the behaviour of solutions to <span>(Lu=0,)</span> as we tend to the boundary along radii in strictly starlike domains in <span>(mathbb {R}^n, nge 3.)</span> Analogous results are obtained in other special domains. Our approach involves introducing harmonic line bundles as instances of Brelot harmonic spaces and approximating continuous functions by harmonic functions on appropriate subsets. We are required to approximate on certain closed sets, which is not obvious, since the space of continuous functions on an (unbounded) closed set, endowed with the topology of uniform convergence, is not a topological vector space, though it is both a vector space and a topological space.\u0000</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"375 - 402"},"PeriodicalIF":0.4,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223714","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":"On (Lambda )-submodules with finite index of the plus/minus Selmer group over anticyclotomic ({{,mathrm{mathbb {Z}},}}_{p})-extension at inert primes","authors":"Ryota Shii","doi":"10.1007/s40316-024-00236-5","DOIUrl":"10.1007/s40316-024-00236-5","url":null,"abstract":"<div><p>Let <i>K</i> be an imaginary quadratic field where a prime number <span>(p ge 5)</span> is inert. Let <i>E</i> be an elliptic curve defined over <i>K</i> and suppose that <i>E</i> has good supersingular reduction at <i>p</i>. In this paper, we prove that the plus/minus Selmer group of <i>E</i> over the anticyclotomic <span>({{,mathrm{mathbb {Z}},}}_{p})</span>-extension of <i>K</i> has no proper <span>(Lambda )</span>-submodules of finite index under mild assumptions for <i>E</i>. This is an analogous result to R. Greenberg and B. D. Kim for the anticyclotomic <span>({{,mathrm{mathbb {Z}},}}_{p})</span>-extension essentially. By applying the results of A. Agboola–B. Howard or A. Burungale–K. Büyükboduk–A. Lei, we can also construct examples satisfying the assumptions of our theorem.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"477 - 490"},"PeriodicalIF":0.4,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223715","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":"Uniqueness of Lagrangians in (T^*{mathbb {R}}P^2)","authors":"Nikolas Adaloglou","doi":"10.1007/s40316-024-00238-3","DOIUrl":"10.1007/s40316-024-00238-3","url":null,"abstract":"<div><p>We present a new and simpler proof of the fact that any Lagrangian <span>({mathbb {R}}P^2)</span> in <span>(T^*{mathbb {R}}P^2)</span> is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians in <span>(T^*S^2)</span>, using surgery to turn Lagrangian spheres into symplectic ones. The main novel contribution is a detailed proof of the folklore fact that the complement of a symplectic quadric in <span>({mathbb {C}}P^2)</span> can be identified with the unit cotangent disc bundle of <span>({mathbb {R}}P^2)</span>.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 1","pages":"215 - 222"},"PeriodicalIF":0.5,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848947","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}
Thomas P. Lambert, John G. Ratcliffe, Steven T. Tschantz
{"title":"Closed flat Riemannian 4-manifolds","authors":"Thomas P. Lambert, John G. Ratcliffe, Steven T. Tschantz","doi":"10.1007/s40316-024-00231-w","DOIUrl":"10.1007/s40316-024-00231-w","url":null,"abstract":"<div><p>In this paper, we describe the classification of all the geometric fibrations of a closed flat Riemannian 4-manifold over a connected 1-orbifold.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"287 - 333"},"PeriodicalIF":0.4,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-024-00231-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223718","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":"Some remarks on critical sets of Laplace eigenfunctions","authors":"Chris Judge, Sugata Mondal","doi":"10.1007/s40316-024-00240-9","DOIUrl":"10.1007/s40316-024-00240-9","url":null,"abstract":"<p>We study the set of critical points of a solution to <span>(Delta u = lambda cdot u)</span> and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon <i>P</i> has infinitely many critical points, then <i>P</i> is a rectangle.</p>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 1","pages":"155 - 163"},"PeriodicalIF":0.5,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848944","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":"Generators for the moduli space of parabolic bundle","authors":"Lisa Jeffrey, Yukai Zhang","doi":"10.1007/s40316-024-00232-9","DOIUrl":"10.1007/s40316-024-00232-9","url":null,"abstract":"<div><p>The purpose of this note is to find explicit representatives in de Rham cohomology for the generators of the cohomology of the moduli space of parabolic bundles, analogous to the results of [5] for the moduli space of vector bundles. Further we use the explicit generators to compute the intersection pairing of its cohomology.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 1","pages":"223 - 236"},"PeriodicalIF":0.5,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848948","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 heat kernel on curvilinear polygonal domains in surfaces","authors":"Medet Nursultanov, Julie Rowlett, David Sher","doi":"10.1007/s40316-024-00237-4","DOIUrl":"10.1007/s40316-024-00237-4","url":null,"abstract":"<p>We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic expansion of the heat trace and apply this expansion to demonstrate a collection of results showing that corners are spectral invariants.</p>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 1","pages":"1 - 61"},"PeriodicalIF":0.5,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-024-00237-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848952","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}
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