Pub Date : 2022-04-27DOI: 10.4310/cntp.2022.v16.n2.a3
Satoshi Kondo, Taizan Watari
For an elliptic curve $E$ over an abelian extension $k/K$ with CM by $K$ of Shimura type, the L-functions of its $[k:K]$ Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to $E$ pulls back the $1$-forms on $E$ to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with $[E]_mathbb{C}$ ($E$ as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kähler parameter of the target space $[E]_mathbb{C}$ in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.
对于具有CM × k的Shimura型椭圆曲线$E$在阿贝尔扩展$k/ k $上，其$[k: k]$伽罗瓦表示的l -函数是Hecke函数的Mellin变换;从模曲线到$E$的模参数化(满射映射)将$E$上的$1$-形式拉回以得到Hecke函数。本文对前人的研究进行了改进，证明了一类以$[E]_mathbb{C}$ ($E$为实解析流形)为目标空间的II型弦理论中的手性相关函数与模曲线上的对象产生相同的Hecke函数。弦理论中目标空间$[E]_mathbb{C}$的Kähler参数在定义模曲线的射影/直极限时起着索引(偏序)集的作用。
{"title":"Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces","authors":"Satoshi Kondo, Taizan Watari","doi":"10.4310/cntp.2022.v16.n2.a3","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n2.a3","url":null,"abstract":"For an elliptic curve $E$ over an abelian extension $k/K$ with CM by $K$ of Shimura type, the L-functions of its $[k:K]$ Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to $E$ pulls back the $1$-forms on $E$ to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with $[E]_mathbb{C}$ ($E$ as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kähler parameter of the target space $[E]_mathbb{C}$ in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"39 2","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-27DOI: 10.4310/cntp.2022.v16.n2.a1
Shinobu Hosono, Hiromichi Takagi
We study mirror symmetry of a family of Calabi–Yau manifolds fibered by $(1,8)$-polarized abelian surfaces with Euler characteristic zero. By describing the parameter space globally, we find all expected boundary points (LCSLs), including those correspond to Fourier–Mukai partners. Applying mirror symmetry at each boundary point, we calculate Gromov–Witten invariants $(g leq 2)$ and observe nice (quasi-)modular properties in their potential functions. We also describe degenerations of Calabi–Yau manifolds over each boundary point.
{"title":"Mirror symmetry of Calabi-Yau manifolds fibered by $(1,8)$-polarized abelian surfaces","authors":"Shinobu Hosono, Hiromichi Takagi","doi":"10.4310/cntp.2022.v16.n2.a1","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n2.a1","url":null,"abstract":"We study mirror symmetry of a family of Calabi–Yau manifolds fibered by $(1,8)$-polarized abelian surfaces with Euler characteristic zero. By describing the parameter space globally, we find all expected boundary points (LCSLs), including those correspond to Fourier–Mukai partners. Applying mirror symmetry at each boundary point, we calculate Gromov–Witten invariants $(g leq 2)$ and observe nice (quasi-)modular properties in their potential functions. We also describe degenerations of Calabi–Yau manifolds over each boundary point.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"38 2","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-25DOI: 10.4310/cntp.2023.v17.n2.a5
Salvatore Baldino, R. Schiappa, M. Schwick, Roberto Vega
Resurgent-transseries solutions to Painleve equations may be recursively constructed out of these nonlinear differential-equations -- but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painleve I and Painleve II contexts, nonlinear Stokes data get further hindered as these equations are resonant, with non-trivial consequences for the interconnections between transseries sectors, bridge equations, and associated Stokes coefficients. In parallel to this, the Painleve I and Painleve II equations are string-equations for two-dimensional quantum (super) gravity and minimal string theories, where Stokes data have natural ZZ-brane interpretations. This work computes for the first time the complete, analytical, resurgent Stokes data for the first two Painleve equations, alongside their quantum gravity or minimal string incarnations. The method developed herein, dubbed"closed-form asymptotics", makes sole use of resurgent large-order asymptotics of transseries solutions -- alongside a careful analysis of the role resonance plays. Given its generality, it may be applicable to other distinct (nonlinear, resonant) problems. Results for analytical Stokes coefficients have natural structures, which are described, and extensive high-precision numerical tests corroborate all analytical predictions. Connection-formulae are explicitly constructed, with rather simple and compact final results encoding the full Stokes data, and further allowing for exact monodromy checks -- hence for an analytical proof of our results.
{"title":"Resurgent Stokes data for Painlevé equations and two-dimensional quantum (super) gravity","authors":"Salvatore Baldino, R. Schiappa, M. Schwick, Roberto Vega","doi":"10.4310/cntp.2023.v17.n2.a5","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n2.a5","url":null,"abstract":"Resurgent-transseries solutions to Painleve equations may be recursively constructed out of these nonlinear differential-equations -- but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painleve I and Painleve II contexts, nonlinear Stokes data get further hindered as these equations are resonant, with non-trivial consequences for the interconnections between transseries sectors, bridge equations, and associated Stokes coefficients. In parallel to this, the Painleve I and Painleve II equations are string-equations for two-dimensional quantum (super) gravity and minimal string theories, where Stokes data have natural ZZ-brane interpretations. This work computes for the first time the complete, analytical, resurgent Stokes data for the first two Painleve equations, alongside their quantum gravity or minimal string incarnations. The method developed herein, dubbed\"closed-form asymptotics\", makes sole use of resurgent large-order asymptotics of transseries solutions -- alongside a careful analysis of the role resonance plays. Given its generality, it may be applicable to other distinct (nonlinear, resonant) problems. Results for analytical Stokes coefficients have natural structures, which are described, and extensive high-precision numerical tests corroborate all analytical predictions. Connection-formulae are explicitly constructed, with rather simple and compact final results encoding the full Stokes data, and further allowing for exact monodromy checks -- hence for an analytical proof of our results.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42713480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-12DOI: 10.4310/CNTP.2023.v17.n1.a5
Ksenia Fedosova, A. Pohl, J. Rowlett
We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
{"title":"Fourier expansions of vector-valued automorphic functions with non-unitary twists","authors":"Ksenia Fedosova, A. Pohl, J. Rowlett","doi":"10.4310/CNTP.2023.v17.n1.a5","DOIUrl":"https://doi.org/10.4310/CNTP.2023.v17.n1.a5","url":null,"abstract":"We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-05DOI: 10.4310/cntp.2023.v17.n2.a3
A. Brini, Yannik Schuler
We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact log Gromov-Witten invariants of Looijenga pairs to other curve counting invariants of Gromov-Witten/Gopakumar-Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the log invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.
{"title":"On quasi-tame Looijenga pairs","authors":"A. Brini, Yannik Schuler","doi":"10.4310/cntp.2023.v17.n2.a3","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n2.a3","url":null,"abstract":"We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact log Gromov-Witten invariants of Looijenga pairs to other curve counting invariants of Gromov-Witten/Gopakumar-Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the log invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46316385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-03DOI: 10.4310/CNTP.2022.v16.n3.a1
J. Cardy
: Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be preserved under the T T deformation. The formulation and proof of this statement in fact extents to more general functions such as T T deformed modular and Jacobi forms. We show that the deformation acts simply on their Mellin transform, multiplying it by a universal entire function. Finally we show that Maass forms on the torus are eigenfunctions of the T T deformation.
{"title":"$T overline{T}$-deformed modular forms","authors":"J. Cardy","doi":"10.4310/CNTP.2022.v16.n3.a1","DOIUrl":"https://doi.org/10.4310/CNTP.2022.v16.n3.a1","url":null,"abstract":": Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be preserved under the T T deformation. The formulation and proof of this statement in fact extents to more general functions such as T T deformed modular and Jacobi forms. We show that the deformation acts simply on their Mellin transform, multiplying it by a universal entire function. Finally we show that Maass forms on the torus are eigenfunctions of the T T deformation.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44335694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/cntp.2022.v16.n1.a-karayayla
Tolga Karayayla
{"title":"On a class of non-simply connected Calabi-Yau $3$-folds with positive Euler characteristic","authors":"Tolga Karayayla","doi":"10.4310/cntp.2022.v16.n1.a-karayayla","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n1.a-karayayla","url":null,"abstract":"","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-16DOI: 10.4310/CNTP.2022.v16.n3.a2
Andreas Malmendier, Michael T. Schultz
. We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank ρ ≥ 16. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard-Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. We show that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hypersurfaces we show that the mixed-twist construction produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin-Barnes integrals exists whose monodromy we compute explicitly.
{"title":"On the mixed-twist construction and monodromy of associated Picard–Fuchs systems","authors":"Andreas Malmendier, Michael T. Schultz","doi":"10.4310/CNTP.2022.v16.n3.a2","DOIUrl":"https://doi.org/10.4310/CNTP.2022.v16.n3.a2","url":null,"abstract":". We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank ρ ≥ 16. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard-Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. We show that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hypersurfaces we show that the mixed-twist construction produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin-Barnes integrals exists whose monodromy we compute explicitly.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44667130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-12DOI: 10.4310/cntp.2023.v17.n2.a1
Reinier Kramer
We generalise a result of Kazarian regarding Kadomtsev-Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitz-type counting problems or hypergeometric tau-functions. The proof uses recent results on the relations between hypergeometric tau-functions and topological recursion, as well as the Eynard-DOSS correspondence between topological recursion and cohomological field theories. In particular, we recover the result of Alexandrov of KP integrability for triple Hodge integrals with a Calabi-Yau condition.
{"title":"KP hierarchy for Hurwitz-type cohomological field theories","authors":"Reinier Kramer","doi":"10.4310/cntp.2023.v17.n2.a1","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n2.a1","url":null,"abstract":"We generalise a result of Kazarian regarding Kadomtsev-Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitz-type counting problems or hypergeometric tau-functions. The proof uses recent results on the relations between hypergeometric tau-functions and topological recursion, as well as the Eynard-DOSS correspondence between topological recursion and cohomological field theories. In particular, we recover the result of Alexandrov of KP integrability for triple Hodge integrals with a Calabi-Yau condition.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41681649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-16DOI: 10.4310/CNTP.2022.v16.n3.a4
Cid Reyes-Bustos, M. Wakayama
The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias (ibQRM (cid:96) ) was uncovered in recent studies by the explicit construction of operators J (cid:96) commuting with the Hamiltonian. The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, the crossings on the energy curves. In this paper we propose a conjectural relation between the symmetry and degeneracy for the ibQRM (cid:96) given explicitly in terms of two polynomials appearing independently in the respective investigations. Concretely, one of the polynomials appears as the quotient of the constraint polynomials that assure the existence of degenerate solutions while the other determines a quadratic relation (in general, it defines a curve of hyperelliptic type) between the ibQRM (cid:96) Hamiltonian and its basic commuting operator J (cid:96) . Following this conjecture, we derive several interesting structural insights of the whole spectrum. For instance, the energy curves are naturally shown to lie on a surface determined by the family of hyperelliptic curves by considering the coupling constant as a variable. This geometric picture contains the generalization of the parity decomposition of the symmetric quantum Rabi model. Moreover, it allows us to describe a remarkable approximation of the first (cid:96) energy curves by the zero-section of the corresponding hyperelliptic curve. These investigations naturally lead to a geometric picture of the (hyper-)elliptic surfaces given by the Kodaira-N´eron type model for a family of curves over the projective line in connection with the energy curves, which may be expected to provide a complex analytic proof of the conjecture.
{"title":"Degeneracy and hidden symmetry for the asymmetric quantum Rabi model with integral bias","authors":"Cid Reyes-Bustos, M. Wakayama","doi":"10.4310/CNTP.2022.v16.n3.a4","DOIUrl":"https://doi.org/10.4310/CNTP.2022.v16.n3.a4","url":null,"abstract":"The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias (ibQRM (cid:96) ) was uncovered in recent studies by the explicit construction of operators J (cid:96) commuting with the Hamiltonian. The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, the crossings on the energy curves. In this paper we propose a conjectural relation between the symmetry and degeneracy for the ibQRM (cid:96) given explicitly in terms of two polynomials appearing independently in the respective investigations. Concretely, one of the polynomials appears as the quotient of the constraint polynomials that assure the existence of degenerate solutions while the other determines a quadratic relation (in general, it defines a curve of hyperelliptic type) between the ibQRM (cid:96) Hamiltonian and its basic commuting operator J (cid:96) . Following this conjecture, we derive several interesting structural insights of the whole spectrum. For instance, the energy curves are naturally shown to lie on a surface determined by the family of hyperelliptic curves by considering the coupling constant as a variable. This geometric picture contains the generalization of the parity decomposition of the symmetric quantum Rabi model. Moreover, it allows us to describe a remarkable approximation of the first (cid:96) energy curves by the zero-section of the corresponding hyperelliptic curve. These investigations naturally lead to a geometric picture of the (hyper-)elliptic surfaces given by the Kodaira-N´eron type model for a family of curves over the projective line in connection with the energy curves, which may be expected to provide a complex analytic proof of the conjecture.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42334478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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