Mathematische Annalen最新文献

{"title":"Almost positive links are strongly quasipositive.","authors":"Peter Feller,&nbsp;Lukas Lewark,&nbsp;Andrew Lobb","doi":"10.1007/s00208-021-02328-x","DOIUrl":"https://doi.org/10.1007/s00208-021-02328-x","url":null,"abstract":"<p><p>We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give a simple and complete characterization of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"385 1-2","pages":"481-510"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9889511/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9205359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums.","authors":"Paolo Minelli,&nbsp;Athanasios Sourmelidis,&nbsp;Marc Technau","doi":"10.1007/s00208-022-02452-2","DOIUrl":"10.1007/s00208-022-02452-2","url":null,"abstract":"<p><p>We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval (0, 1/2), establishing that they behave differently on (0, 1/2) than they do on (1/2, 1). These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovskiĭ and others, ultimately dating back to Lochs and Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"387 1-2","pages":"291-320"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10442276/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10056888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the monodromy of the deformed cubic oscillator.","authors":"Tom Bridgeland,&nbsp;Davide Masoero","doi":"10.1007/s00208-021-02337-w","DOIUrl":"https://doi.org/10.1007/s00208-021-02337-w","url":null,"abstract":"<p><p>We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Riemann-Hilbert problems of (Bridgeland in Invent Math 216(1):69-124, 2019) arising from the Donaldson-Thomas theory of the A <math><msub><mrow></mrow> <mn>2</mn></msub> </math> quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"385 1-2","pages":"193-258"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9889533/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10708173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Green function estimates on complements of low-dimensional uniformly rectifiable sets.","authors":"Guy David,&nbsp;Joseph Feneuil,&nbsp;Svitlana Mayboroda","doi":"10.1007/s00208-022-02379-8","DOIUrl":"https://doi.org/10.1007/s00208-022-02379-8","url":null,"abstract":"<p><p>It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions. arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the \"flagship\" degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators <math> <mrow><msub><mi>L</mi> <mrow><mi>β</mi> <mo>,</mo> <mi>γ</mi></mrow> </msub> <mo>=</mo> <mo>-</mo> <mtext>div</mtext> <msup><mi>D</mi> <mrow><mi>d</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>γ</mi> <mo>-</mo> <mi>n</mi></mrow> </msup> <mi>∇</mi></mrow> </math> associated to a domain <math><mrow><mi>Ω</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> </mrow> </math> with a uniformly rectifiable boundary <math><mi>Γ</mi></math> of dimension <math><mrow><mi>d</mi> <mo><</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn></mrow> </math> , the now usual distance to the boundary <math><mrow><mi>D</mi> <mo>=</mo> <msub><mi>D</mi> <mi>β</mi></msub> </mrow> </math> given by <math> <mrow><msub><mi>D</mi> <mi>β</mi></msub> <msup><mrow><mo>(</mo> <mi>X</mi> <mo>)</mo></mrow> <mrow><mo>-</mo> <mi>β</mi></mrow> </msup> <mo>=</mo> <msub><mo>∫</mo> <mi>Γ</mi></msub> <msup><mrow><mo>|</mo> <mi>X</mi> <mo>-</mo> <mi>y</mi> <mo>|</mo></mrow> <mrow><mo>-</mo> <mi>d</mi> <mo>-</mo> <mi>β</mi></mrow> </msup> <mi>d</mi> <mi>σ</mi> <mrow><mo>(</mo> <mi>y</mi> <mo>)</mo></mrow> </mrow> </math> for <math><mrow><mi>X</mi> <mo>∈</mo> <mi>Ω</mi></mrow> </math> , where <math><mrow><mi>β</mi> <mo>></mo> <mn>0</mn></mrow> </math> and <math><mrow><mi>γ</mi> <mo>∈</mo> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo></mrow> </math> . In this paper we show that the Green function <i>G</i> for <math><msub><mi>L</mi> <mrow><mi>β</mi> <mo>,</mo> <mi>γ</mi></mrow> </msub> </math> , with pole at infinity, is well approximated by multiples of <math><msup><mi>D</mi> <mrow><mn>1</mn> <mo>-</mo> <mi>γ</mi></mrow> </msup> </math> , in the sense that the function <math> <mrow><mrow><mo>|</mo></mrow> <mi>D</mi> <mi>∇</mi> <mrow><mo>(</mo></mrow> <mo>ln</mo> <mrow><mo>(</mo></mrow> <mfrac><mi>G</mi> <msup><mi>D</mi> <mrow><mn>1</mn> <mo>-</mo> <mi>γ</mi></mrow> </msup> </mfrac> <mrow><mo>)</mo></mrow> <mrow><mo>)</mo></mrow> <msup><mrow><mo>|</mo></mrow> <mn>2</mn></msup> </mrow> </math> satisfies a Carleson measure estimate on <math><mi>Ω</mi></math> . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the \"magical\" distance function from David et al. (Duke Math J, to appear).</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"385 3-4","pages":"1797-1821"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10042934/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9603398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The space of homogeneous probability measures on <ns0:math><ns0:msubsup><ns0:mover><ns0:mrow><ns0:mi>Γ</ns0:mi><ns0:mo></ns0:mo><ns0:mi>X</ns0:mi></ns0:mrow><ns0:mo>¯</ns0:mo></ns0:mover><ns0:mi>max</ns0:mi><ns0:mi>S</ns0:mi></ns0:msubsup></ns0:math> is compact: With an appendix by Jialun Li.","authors":"Christopher Daw,&nbsp;Alexander Gorodnik,&nbsp;Emmanuel Ullmo","doi":"10.1007/s00208-022-02412-w","DOIUrl":"https://doi.org/10.1007/s00208-022-02412-w","url":null,"abstract":"<p><p>In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"386 1-2","pages":"987-1016"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10163147/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10300601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arithmetic statistics of Prym surfaces.","authors":"Jef Laga","doi":"10.1007/s00208-022-02398-5","DOIUrl":"https://doi.org/10.1007/s00208-022-02398-5","url":null,"abstract":"<p><p>We consider a family of abelian surfaces over <math><mi>Q</mi></math> arising as Prym varieties of double covers of genus-1 curves by genus-3 curves. These abelian surfaces carry a polarization of type (1, 2) and we show that the average size of the Selmer group of this polarization equals 3. Moreover we show that the average size of the 2-Selmer group of the abelian surfaces in the same family is bounded above by 5. This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not principally polarized. The proof is a combination of an analysis of the Lie algebra embedding <math><mrow><msub><mi>F</mi><mn>4</mn></msub><mo>⊂</mo><msub><mi>E</mi><mn>6</mn></msub></mrow></math>, invariant theory, a classical geometric construction due to Pantazis, a study of Néron component groups of Prym surfaces and Bhargava's orbit-counting techniques.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"386 1-2","pages":"247-327"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10163148/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10300600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum.","authors":"Arthur Bik, Alessandro Danelon, Jan Draisma","doi":"10.1007/s00208-022-02386-9","DOIUrl":"10.1007/s00208-022-02386-9","url":null,"abstract":"<p><p>In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free <i>R</i>-modules to finitely generated <i>R</i>-modules, for any commutative ring <i>R</i> whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with <math><mrow><mi>R</mi> <mo>=</mo> <mrow><mspace></mspace> <mi>Z</mi> <mspace></mspace></mrow> </mrow> </math> to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated <i>R</i>-module <i>M</i> we associate a topological space, which we show is Noetherian when <math> <mrow><mrow><mspace></mspace> <mi>Spec</mi> <mspace></mspace></mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo></mrow> </math> is; this is the degree-zero case of our result on polynomial functors.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"385 3-4","pages":"1879-1921"},"PeriodicalIF":1.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10042986/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9603397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}