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Expositiones Mathematicae最新文献

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On Bott–Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms 几乎复流形的bot - chern和Aeppli上同调及调和形式的相关空间
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-14 DOI: 10.1016/j.exmath.2023.09.001
Lorenzo Sillari , Adriano Tomassini
{"title":"On Bott–Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms","authors":"Lorenzo Sillari ,&nbsp;Adriano Tomassini","doi":"10.1016/j.exmath.2023.09.001","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.09.001","url":null,"abstract":"<div><p>In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott–Chern and Aeppli cohomologies defined using the operators <span><math><mi>d</mi></math></span>, <span><math><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to <span><math><mi>d</mi></math></span>, <span><math><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>, showing their relation with Bott–Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott–Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an almost complex invariant <span><math><msubsup><mrow><mi>h</mi></mrow><mrow><mi>d</mi><mo>+</mo><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> that distinguishes between almost complex structures. On almost Kähler 4-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49863758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Splitting fields of Xn−X−1 (particularly for n=5), prime decomposition and modular forms 分割Xn−X−1的域(特别是当n=5时)，素数分解和模形式
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.02.007
Chandrashekhar B. Khare , Alfio Fabio La Rosa , Gabor Wiese
{"title":"Splitting fields of Xn−X−1 (particularly for n=5), prime decomposition and modular forms","authors":"Chandrashekhar B. Khare ,&nbsp;Alfio Fabio La Rosa ,&nbsp;Gabor Wiese","doi":"10.1016/j.exmath.2023.02.007","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.007","url":null,"abstract":"<div><p><span>We study the splitting fields of the family of polynomials </span><span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>X</mi><mo>−</mo><mn>1</mn></mrow></math></span>. This family of polynomials has been much studied in the literature and has some remarkable properties. In Serre (2003), Serre related the function on primes <span><math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, for a fixed <span><math><mrow><mi>n</mi><mo>≤</mo><mn>4</mn></mrow></math></span> and <span><math><mi>p</mi></math></span> a varying prime, which counts the number of roots of <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> to coefficients of modular forms. We study the case <span><math><mrow><mi>n</mi><mo>=</mo><mn>5</mn></mrow></math></span>, and relate <span><math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> to mod 5 modular forms over <span><math><mi>Q</mi></math></span>, and to characteristic 0, parallel weight 1 Hilbert modular forms over <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mn>19</mn><mi>⋅</mi><mn>151</mn></mrow></msqrt><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49865975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Geometric quadratic Chabauty and p-adic heights 几何二次恰布蒂和p进高度
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.05.003
Juanita Duque-Rosero , Sachi Hashimoto , Pim Spelier
{"title":"Geometric quadratic Chabauty and p-adic heights","authors":"Juanita Duque-Rosero ,&nbsp;Sachi Hashimoto ,&nbsp;Pim Spelier","doi":"10.1016/j.exmath.2023.05.003","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.05.003","url":null,"abstract":"<div><p>Let <span><math><mi>X</mi></math></span> be a curve of genus <span><math><mrow><mi>g</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span> over <span><math><mi>Q</mi></math></span> whose Jacobian <span><math><mi>J</mi></math></span> has Mordell–Weil rank <span><math><mi>r</mi></math></span> and Néron–Severi rank <span><math><mi>ρ</mi></math></span>. When <span><math><mrow><mi>r</mi><mo>&lt;</mo><mi>g</mi><mo>+</mo><mi>ρ</mi><mo>−</mo><mn>1</mn></mrow></math></span>, the geometric quadratic Chabauty method determines a finite set of <span><math><mi>p</mi></math></span>-adic points containing the rational points of <span><math><mi>X</mi></math></span>. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of <span><math><mi>p</mi></math></span>-adic heights and <span><math><mi>p</mi></math></span>-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of <span><math><mi>p</mi></math></span>-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49865972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Degeneration locus of Qp-local systems: Conjectures qp -局部系统的退化轨迹:猜想
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.05.002
Anna Cadoret
{"title":"Degeneration locus of Qp-local systems: Conjectures","authors":"Anna Cadoret","doi":"10.1016/j.exmath.2023.05.002","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.05.002","url":null,"abstract":"<div><p>We introduce a conjecture on the arithmetic sparcity of the degeneration locus of a <span><math><mi>p</mi></math></span>-adic local system on a smooth variety over a number field and, <span><em>modulo</em></span> the Bombieri–Lang conjecture, show that it follows from a conjecture on the geometry of the level varieties attached to the local system. We present a few applications of our conjecture to classical problems in arithmetic geometry. Eventually, we give some evidences and discuss a few perspectives to attack it, in particular for <span><math><mi>p</mi></math></span>-adic local systems arising from geometry.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49865973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Logarithmic moduli of roots of line bundles on curves 曲线上线束根的对数模
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.04.001
David Holmes , Giulio Orecchia
{"title":"Logarithmic moduli of roots of line bundles on curves","authors":"David Holmes ,&nbsp;Giulio Orecchia","doi":"10.1016/j.exmath.2023.04.001","DOIUrl":"10.1016/j.exmath.2023.04.001","url":null,"abstract":"<div><p>We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a ‘double ramification cycle’ measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho–Pandharipande–Schmitt and Holmes–Schwarz).</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48520614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
A K3 surface related to Leonardo Pisano’s work on congruent numbers 与Leonardo Pisano关于全等数的研究有关的K3曲面
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.03.003
Martin Djukanović, Jaap Top
{"title":"A K3 surface related to Leonardo Pisano’s work on congruent numbers","authors":"Martin Djukanović,&nbsp;Jaap Top","doi":"10.1016/j.exmath.2023.03.003","DOIUrl":"10.1016/j.exmath.2023.03.003","url":null,"abstract":"<div><p>This note recalls an early 13th century result on congruent numbers by Leonardo Pisano (“Fibonacci”), and shows how it relates to a specific much studied K3 surface and to an elliptic fibration on this surface. As an aside, the discussion reveals how, via explicit maps of degree two, the surface is covered by the Fermat quartic surface and also covers one of the two famous ‘most algebraic K3 surfaces’.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41595977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extensions and torsors for finite group schemes 有限群格式的扩张与扭转
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.02.004
Peter Bruin
{"title":"Extensions and torsors for finite group schemes","authors":"Peter Bruin","doi":"10.1016/j.exmath.2023.02.004","DOIUrl":"10.1016/j.exmath.2023.02.004","url":null,"abstract":"<div><p>We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite locally free commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43076278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
An overview on problems of Unlikely Intersections in families of abelian varieties 阿贝尔变种族中不可能相交问题综述
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.04.003
Laura Capuano
{"title":"An overview on problems of Unlikely Intersections in families of abelian varieties","authors":"Laura Capuano","doi":"10.1016/j.exmath.2023.04.003","DOIUrl":"10.1016/j.exmath.2023.04.003","url":null,"abstract":"<div><p>This short survey is part of a minicourse I gave during the CMI-HIMR Summer School “Unlikely Intersections in Diophantine Geometry” on the Zilber–Pink conjecture, formulated independently by Zilber (2002), Bombieri, Masser and Zannier (1999) in the case of tori and by Pink (2005) in the more general setting of mixed Shimura varieties. This conjecture, which includes in its general formulation many important results in number theory<span>, has been intensively studied by several mathematicians in the past 20 years. We will mainly focus on these problems in the special setting of semiabelian varieties and families of abelian varieties.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42436947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Computing the trace of an algebraic point on an elliptic curve 计算椭圆曲线上代数点的轨迹
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.02.006
Nicolas Mascot , Denis Simon
{"title":"Computing the trace of an algebraic point on an elliptic curve","authors":"Nicolas Mascot ,&nbsp;Denis Simon","doi":"10.1016/j.exmath.2023.02.006","DOIUrl":"10.1016/j.exmath.2023.02.006","url":null,"abstract":"<div><p>We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43843514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Problèmes de type André-Oort en pinceau arithmétique 算术笔画中的andre - oort问题
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-09-01 DOI: 10.1016/j.exmath.2023.05.004
Rodolphe Richard
{"title":"Problèmes de type André-Oort en pinceau arithmétique","authors":"Rodolphe Richard","doi":"10.1016/j.exmath.2023.05.004","DOIUrl":"10.1016/j.exmath.2023.05.004","url":null,"abstract":"<div><p>Nous proposons une “Conjecture d’André-Oort en pinceau arithmétique”.</p><p>C’est une extension de la conjecture d’André-Oort, disons “classique”, formulée à l’origine par Y. André et F. Oort. La conjecture fait intervenir les modèles entiers des variétés de Shimura.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43293945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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