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Mukai models and Borcherds products 向井模型和 Borcherds 产品
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928323
Shouhei Ma
{"title":"Mukai models and Borcherds products","authors":"Shouhei Ma","doi":"10.1353/ajm.2024.a928323","DOIUrl":"https://doi.org/10.1353/ajm.2024.a928323","url":null,"abstract":"<p><p>abstract:</p><p>Let ${Fgn}$ be the moduli space of $n$-pointed $K3$ surfaces of genus $g$ with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on ${Fgn}$ and the ring of certain orthogonal modular forms, and give applications to the birational type of ${Fgn}$. We prove that the Kodaira dimension of ${Fgn}$ stabilizes to $19$ when $n$ is sufficiently large. Then we use explicit Borcherds products to find a lower bound of $n$ where ${Fgn}$ has nonnegative Kodaira dimension, and compare this with an upper bound where ${Fgn}$ is unirational or uniruled using Mukai models of $K3$ surfaces in $gleq 20$. This reveals the exact transition point of Kodaira dimension in some~$g$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147427","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}
引用次数: 0
Character rigidity for lattices and commensurators 网格和换元器的特征刚性
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928322
Darren Creutz, Jesse Peterson
{"title":"Character rigidity for lattices and commensurators","authors":"Darren Creutz, Jesse Peterson","doi":"10.1353/ajm.2024.a928322","DOIUrl":"https://doi.org/10.1353/ajm.2024.a928322","url":null,"abstract":"<p><p>abstract:</p><p>We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147412","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}
引用次数: 0
Howe correspondence of unipotent characters for a finite symplectic/even-orthogonal dual pair 有限交映/偶正交对偶的单偶性字符的豪对应关系
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928326
Shu-Yen Pan
{"title":"Howe correspondence of unipotent characters for a finite symplectic/even-orthogonal dual pair","authors":"Shu-Yen Pan","doi":"10.1353/ajm.2024.a928326","DOIUrl":"https://doi.org/10.1353/ajm.2024.a928326","url":null,"abstract":"<p><p>abstract:</p><p>In this paper we give a complete and explicit description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization. That is, the conjecture by Aubert-Michel-Rouquier is confirmed.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147530","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}
引用次数: 0
Stable cones in the thin one-phase problem 薄单相问题中的稳定锥体
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928321
Xavier Fernández-Real, Xavier Ros-Oton
{"title":"Stable cones in the thin one-phase problem","authors":"Xavier Fernández-Real, Xavier Ros-Oton","doi":"10.1353/ajm.2024.a928321","DOIUrl":"https://doi.org/10.1353/ajm.2024.a928321","url":null,"abstract":"<p><p>abstract:</p><p>The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem.</p><p>The problem of classifying stable (or minimal) homogeneous solutions in dimensions $ngeq3$ is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open.</p><p>The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of ``large solutions'' for the fractional Laplacian, which blow up on the free boundary.</p><p>On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions $nle 5$ is one-dimensional, emph{independently} of the parameter $sin (0,1)$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147420","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}
引用次数: 0
Number of solutions to a special type of unit equations in two unknowns 两个未知数中特殊单元方程的解数
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-03-29 DOI: 10.1353/ajm.2024.a923236
Takafumi Miyazaki, István Pink
{"title":"Number of solutions to a special type of unit equations in two unknowns","authors":"Takafumi Miyazaki, István Pink","doi":"10.1353/ajm.2024.a923236","DOIUrl":"https://doi.org/10.1353/ajm.2024.a923236","url":null,"abstract":"<p><p>abstract:</p><p>For any fixed relatively prime positive integers $a$, $b$ and $c$ with $min{a,b,c}&gt;1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x$, $y$ and $z$, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.~A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), no.~2, 897--922] which asserts that Pillai's type equation $a^x-b^y=c$ has at most two solutions in positive integers $x$ and $y$ for any fixed positive integers $a$, $b$ and $c$ with $min{a,b}&gt;1$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323891","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}
引用次数: 0
Non-abelian p-adic Rankin-Selberg L-functions and non-vanishing of central L-values 非阿贝尔 p-adic Rankin-Selberg L 函数和中心 L 值的非凡性
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-03-29 DOI: 10.1353/ajm.2024.a923241
Fabian Januszewski
{"title":"Non-abelian p-adic Rankin-Selberg L-functions and non-vanishing of central L-values","authors":"Fabian Januszewski","doi":"10.1353/ajm.2024.a923241","DOIUrl":"https://doi.org/10.1353/ajm.2024.a923241","url":null,"abstract":"<p><p>abstract:</p><p>We construct $p$-adic $L$-functions for torsion classes for $GL(n+1)timesGL(n)$ and along the way prove new congruences between special values of Rankin-Selberg $L$-functions for $GL(n+1)timesGL(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $GL(n!+!1)linebreaktimesGL(n)$. As an application, we establish generic non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140324246","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}
引用次数: 0
Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials 硬边平面正交多项式的黎曼-希尔伯特层次结构
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-03-29 DOI: 10.1353/ajm.2024.a923237
Haakan Hedenmalm, Aron Wennman
{"title":"Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials","authors":"Haakan Hedenmalm, Aron Wennman","doi":"10.1353/ajm.2024.a923237","DOIUrl":"https://doi.org/10.1353/ajm.2024.a923237","url":null,"abstract":"<p><p>abstract:</p><p>We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $varkappa$, the expansion holds with an $mathrm{O}(N^{-varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies---sequences of scalar Riemann-Hilbert problems---which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $partialmathscr{D}$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323850","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}
引用次数: 0
Examples of property (T) II1 factors with trivial fundamental group 基本群微不足道的性质 (T) II1 因子实例
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-03-29 DOI: 10.1353/ajm.2024.a923239
Ionuţ Chifan, Sayan Das, Cyril Houdayer, Krishnendu Khan
{"title":"Examples of property (T) II1 factors with trivial fundamental group","authors":"Ionuţ Chifan, Sayan Das, Cyril Houdayer, Krishnendu Khan","doi":"10.1353/ajm.2024.a923239","DOIUrl":"https://doi.org/10.1353/ajm.2024.a923239","url":null,"abstract":"<p><p>abstract:</p><p>In this article we provide the first examples of property (T) $mathrm{II}_1$ factors $mathcal{N}$ with trivial fundamental group, $mathcal{F}(mathcal{N})=1$. Our examples arise as group factors $mathcal{N}=mathcal{L}(G)$ where $G$ belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in [Geom. Dedicata 112 (2005), 183--196] and the ones introduced recently in [Anal. PDE 16 (2023), 433--476] using the Belegradek-Osin Rips construction from [Groups Geom. Dyn. 2 (2008), 1--12]. In particular, our results provide a continuum of explicit pairwise nonisomorphic property (T) factors.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323985","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}
引用次数: 0
A proof of conjectured partition identities of Nandi 南迪分区特性猜想的证明
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-03-29 DOI: 10.1353/ajm.2024.a923238
Motoki Takigiku, Shunsuke Tsuchioka
{"title":"A proof of conjectured partition identities of Nandi","authors":"Motoki Takigiku, Shunsuke Tsuchioka","doi":"10.1353/ajm.2024.a923238","DOIUrl":"https://doi.org/10.1353/ajm.2024.a923238","url":null,"abstract":"<p><p>abstract:</p><p>We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities for modulus 14 that were posed by Nandi through a vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra $A^{(2)}_{2}$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323989","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}
引用次数: 0
Random walks on tori and normal numbers in self-similar sets 自相似集合中的环上随机游走和正态数
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2024-03-29 DOI: 10.1353/ajm.2024.a923240
Yiftach Dayan, Arijit Ganguly, Barak Weiss
{"title":"Random walks on tori and normal numbers in self-similar sets","authors":"Yiftach Dayan, Arijit Ganguly, Barak Weiss","doi":"10.1353/ajm.2024.a923240","DOIUrl":"https://doi.org/10.1353/ajm.2024.a923240","url":null,"abstract":"<p><p>abstract:</p><p>We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $Ksubsetmathbb{R}^d$ is an attractor of a finite iterated function system of $ngeq 2$ maps of the form $xmapsto D^{-1}x+t_i$ ($i=1,dotsc,n$), where $D$ is an expanding $dtimes d$ integer matrix, and is the same for all the maps, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $xmapsto Dx$ (multiplication mod $mathbb{Z}^d$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base $3$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323849","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}
引用次数: 0
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