Santiago Arango-Piñeros, Deewang Bhamidipati, Soumya Sankar
Given a $g$-dimensional abelian variety $A$ over a finite field $mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $ggeq 3$.
给定有限域$mathbf{F}_{q}$上的$g$维无性杂交$A$，韦尔猜想意味着归一化弗罗贝纽斯特征值生成一个秩最多$g$的乘法群。这个群的庞特里亚金对偶群是一个紧凑的非良性李群，它控制着弗罗贝纽斯内态高次幂的分布。我们称这个群为塞雷-弗罗贝尼斯群，它编码了弗罗贝尼斯特征值之间可能存在的乘法关系。在本文中，我们对 $g le 3$ 时可能出现的所有 Serre-Frobenius 群进行了分类。我们还给出了素维 $ggeq 3$ 的简单普通无性变体的部分分类。
{"title":"Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields","authors":"Santiago Arango-Piñeros, Deewang Bhamidipati, Soumya Sankar","doi":"10.1093/imrn/rnae148","DOIUrl":"https://doi.org/10.1093/imrn/rnae148","url":null,"abstract":"Given a $g$-dimensional abelian variety $A$ over a finite field $mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $ggeq 3$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain a vanishing result for solutions of the inequality $|Delta u| leq q_{1} |u| + q_{2} |nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is related to the behavior of the potential functions $q_{1}$ and $q_{2}$ and to the asymptotic geometry of the end. The main ingredient is a new Carleman estimate of independent interest. Geometric applications to conformal deformations and to minimal graphs are presented.
{"title":"Unique Continuation at Infinity: Carleman Estimates on General Warped Cylinders","authors":"Nicolò De Ponti, Stefano Pigola, Giona Veronelli","doi":"10.1093/imrn/rnae147","DOIUrl":"https://doi.org/10.1093/imrn/rnae147","url":null,"abstract":"We obtain a vanishing result for solutions of the inequality $|Delta u| leq q_{1} |u| + q_{2} |nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is related to the behavior of the potential functions $q_{1}$ and $q_{2}$ and to the asymptotic geometry of the end. The main ingredient is a new Carleman estimate of independent interest. Geometric applications to conformal deformations and to minimal graphs are presented.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $lambda $ denote the Liouville function. We show that for all $N geq 11$, the (non-trivial) convolution sum bound $$ begin{align*} & left|sum_{n < N} lambda(n) lambda(N-n)right| < N-1 end{align*} $$ holds. We also determine all $N$ for which no cancellation in the convolution sum occurs. This answers a question posed at the 2018 AIM workshop on Sarnak’s conjecture.
{"title":"On a Goldbach-Type Problem for the Liouville Function","authors":"Alexander P Mangerel","doi":"10.1093/imrn/rnae149","DOIUrl":"https://doi.org/10.1093/imrn/rnae149","url":null,"abstract":"Let $lambda $ denote the Liouville function. We show that for all $N geq 11$, the (non-trivial) convolution sum bound $$ begin{align*} & left|sum_{n &lt; N} lambda(n) lambda(N-n)right| &lt; N-1 end{align*} $$ holds. We also determine all $N$ for which no cancellation in the convolution sum occurs. This answers a question posed at the 2018 AIM workshop on Sarnak’s conjecture.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective ${mathbb{Z}} G$-modules representing the Swan finiteness obstruction. We use this to determine when $X vee S^{n} simeq Y vee S^{n}$ implies $X simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective ${mathbb{Z}} G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise that lead to a new approach to finding a counterexample to Wall’s D2 problem.
在之前的工作中，我们将$G$具有周期同调时有限$(G,n)$复数的同调类型与代表斯旺有限性障碍的投影${/mathbb{Z}}相关联。代表斯旺有限性障碍的 G$ 模块。我们利用这一点来确定当 $X vee S^{n}simeq Y vee S^{n}$ 对于有限的 $(G,n)$ 复数 $X$ 和 $Y$，意味着 $X simeq Y$，并给出了当这种情况失效时同源不同对的数量下限。证明涉及构造投影 ${mathbb{Z}}G$ 模块作为四元数代数乘积阶上局部自由模块的提升，其存在性源于艾希勒质量公式。在 $n=2$ 的情况下，会出现一些困难，从而导致一种新的方法来寻找沃尔 D2 问题的反例。
{"title":"Cancellation for (G,n)-complexes and the Swan Finiteness Obstruction","authors":"John Nicholson","doi":"10.1093/imrn/rnae141","DOIUrl":"https://doi.org/10.1093/imrn/rnae141","url":null,"abstract":"In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective ${mathbb{Z}} G$-modules representing the Swan finiteness obstruction. We use this to determine when $X vee S^{n} simeq Y vee S^{n}$ implies $X simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective ${mathbb{Z}} G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise that lead to a new approach to finding a counterexample to Wall’s D2 problem.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The non-abelian Hodge correspondence maps a polystable $textrm{SL}(2, {mathbb{R}})$-Higgs bundle on a compact Riemann surface $X$ of genus $g geq 2$ to a connection that, in some cases, is the holonomy of a branched hyperbolic structure. Gaiotto’s conformal limit maps the same bundle to a partial oper, that is, to a connection whose holonomy is that of a branched complex projective structure compatible with $X$. In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with $X$. We also show that, for zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichmüller’s space.
{"title":"The Conformal Limit and Projective Structures","authors":"Pedro M Silva, Peter B Gothen","doi":"10.1093/imrn/rnae142","DOIUrl":"https://doi.org/10.1093/imrn/rnae142","url":null,"abstract":"The non-abelian Hodge correspondence maps a polystable $textrm{SL}(2, {mathbb{R}})$-Higgs bundle on a compact Riemann surface $X$ of genus $g geq 2$ to a connection that, in some cases, is the holonomy of a branched hyperbolic structure. Gaiotto’s conformal limit maps the same bundle to a partial oper, that is, to a connection whose holonomy is that of a branched complex projective structure compatible with $X$. In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with $X$. We also show that, for zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichmüller’s space.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a general counting result for arcs of the same type in compact surfaces. We also count infinite arcs in cusped surfaces and arcs in orbifolds. These theorems are derived from a result that ensures the convergence of certain measures on the space of geodesic currents.
{"title":"Counting Arcs of the Same Type","authors":"Marie Trin","doi":"10.1093/imrn/rnae143","DOIUrl":"https://doi.org/10.1093/imrn/rnae143","url":null,"abstract":"We prove a general counting result for arcs of the same type in compact surfaces. We also count infinite arcs in cusped surfaces and arcs in orbifolds. These theorems are derived from a result that ensures the convergence of certain measures on the space of geodesic currents.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For 0-cycles on a variety over a number field, we define an analogue of the classical descent set for rational points. This leads to, among other things, a definition of the étale-Brauer obstruction set for 0-cycles. We show that all these constructions are compatible with Suslin’s singular homology of degree 0. We then transfer some tools and techniques used to study the arithmetic of rational points into the setting of 0-cycles. For example, we extend the strategy developed by Y. Liang, relating the arithmetic of rational points over finite extensions of the base field to that of 0-cycles, to torsors. We give applications of our results to study the arithmetic behaviour of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties, universal torsors, and torsors under tori.
{"title":"Descent and Étale-Brauer Obstructions for 0-Cycles","authors":"Francesca Balestrieri, Jennifer Berg","doi":"10.1093/imrn/rnae140","DOIUrl":"https://doi.org/10.1093/imrn/rnae140","url":null,"abstract":"For 0-cycles on a variety over a number field, we define an analogue of the classical descent set for rational points. This leads to, among other things, a definition of the étale-Brauer obstruction set for 0-cycles. We show that all these constructions are compatible with Suslin’s singular homology of degree 0. We then transfer some tools and techniques used to study the arithmetic of rational points into the setting of 0-cycles. For example, we extend the strategy developed by Y. Liang, relating the arithmetic of rational points over finite extensions of the base field to that of 0-cycles, to torsors. We give applications of our results to study the arithmetic behaviour of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties, universal torsors, and torsors under tori.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Adam Kanigowski, Mariusz Lemańczyk, Florian K Richter, Joni Teräväinen
We consider vanishing properties of exponential sums of the Liouville function $boldsymbol{lambda }$ of the form $$ begin{align*} & lim_{Htoinfty}limsup_{Xtoinfty}frac{1}{log X}sum_{mleq X}frac{1}{m}sup_{alphain C}bigg|frac{1}{H}sum_{hleq H}boldsymbol{lambda}(m+h)e^{2pi ihalpha}bigg|=0, end{align*} $$ where $Csubset{{mathbb{T}}}$. The case $C={{mathbb{T}}}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $Csubset{{mathbb{T}}}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C={{mathbb{T}}}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2pi ihalpha }$ is replaced by a polynomial phase $e^{2pi ih^{t}alpha }$ for $tgeq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $< 1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of “rigid” sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.
{"title":"On the Local Fourier Uniformity Problem for Small Sets","authors":"Adam Kanigowski, Mariusz Lemańczyk, Florian K Richter, Joni Teräväinen","doi":"10.1093/imrn/rnae134","DOIUrl":"https://doi.org/10.1093/imrn/rnae134","url":null,"abstract":"We consider vanishing properties of exponential sums of the Liouville function $boldsymbol{lambda }$ of the form $$ begin{align*} & lim_{Htoinfty}limsup_{Xtoinfty}frac{1}{log X}sum_{mleq X}frac{1}{m}sup_{alphain C}bigg|frac{1}{H}sum_{hleq H}boldsymbol{lambda}(m+h)e^{2pi ihalpha}bigg|=0, end{align*} $$ where $Csubset{{mathbb{T}}}$. The case $C={{mathbb{T}}}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $Csubset{{mathbb{T}}}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C={{mathbb{T}}}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2pi ihalpha }$ is replaced by a polynomial phase $e^{2pi ih^{t}alpha }$ for $tgeq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $&lt; 1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of “rigid” sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrei Martínez-Finkelshtein, Rafael Morales, Daniel Perales
We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko–Pastur, reciprocal Marchenko–Pastur, and free beta laws, which has an independent interest within free probability.
{"title":"Real Roots of Hypergeometric Polynomials via Finite Free Convolution","authors":"Andrei Martínez-Finkelshtein, Rafael Morales, Daniel Perales","doi":"10.1093/imrn/rnae120","DOIUrl":"https://doi.org/10.1093/imrn/rnae120","url":null,"abstract":"We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko–Pastur, reciprocal Marchenko–Pastur, and free beta laws, which has an independent interest within free probability.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Witt algebra ${mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $textbf{V}_{n}=textbf{C}[x_{1}, ldots , x_{n}]$ (or of algebraic vector fields on $textbf{A}^{n}$). A representation of ${mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $textbf{Fin}^{textrm{op}}$, where $textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.
{"title":"Polynomial Representations of the Witt Lie Algebra","authors":"Steven V Sam, Andrew Snowden, Philip Tosteson","doi":"10.1093/imrn/rnae139","DOIUrl":"https://doi.org/10.1093/imrn/rnae139","url":null,"abstract":"The Witt algebra ${mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $textbf{V}_{n}=textbf{C}[x_{1}, ldots , x_{n}]$ (or of algebraic vector fields on $textbf{A}^{n}$). A representation of ${mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $textbf{Fin}^{textrm{op}}$, where $textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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