Pub Date : 2021-09-07DOI: 10.2140/pjm.2022.317.143
R. Golovko
In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of exact Lagrangian fillings up to Hamiltonian isotopy. They are obtained from the examples of Casals and Ng by applying to them the spherical spinning construction.
{"title":"A note on the infinite number of exact Lagrangian fillings for spherical spuns","authors":"R. Golovko","doi":"10.2140/pjm.2022.317.143","DOIUrl":"https://doi.org/10.2140/pjm.2022.317.143","url":null,"abstract":"In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of exact Lagrangian fillings up to Hamiltonian isotopy. They are obtained from the examples of Casals and Ng by applying to them the spherical spinning construction.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44181021","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 : 2021-08-31DOI: 10.2140/pjm.2021.312.457
F. Giovanni, M. Trombetti
{"title":"Cohopfian groups and accessible group classes","authors":"F. Giovanni, M. Trombetti","doi":"10.2140/pjm.2021.312.457","DOIUrl":"https://doi.org/10.2140/pjm.2021.312.457","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42697804","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 : 2021-08-30DOI: 10.2140/pjm.2023.322.381
Joshua Males, Andreas Mono
By comparing two different evaluations of a modified (`{a} la Borcherds) higher Siegel theta lift on even lattices of signature $(r,s)$, we prove Eichler--Selberg type relations for a wide class of negative weight vector-valued mock modular forms. In doing so, we detail several properties of the lift, as well as showing that it produces an infinite family of local (and locally harmonic) Maa{ss} forms on Grassmanians in certain signatures.
通过比较签名$(r,s)$的偶格上修正的( {a} la Borcherds)较高Siegel theta提升的两种不同的求值，我们证明了一类广泛的负权向量值模拟模形式的Eichler—Selberg型关系。在此过程中，我们详细介绍了升力的几个性质，并证明了它在某些签名的格拉斯曼子上产生无限族的局部(和局部调和)Maa{ss}形式。
{"title":"Local Maass forms and Eichler–Selberg relations\u0000for negative-weight vector-valued mock modular forms","authors":"Joshua Males, Andreas Mono","doi":"10.2140/pjm.2023.322.381","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.381","url":null,"abstract":"By comparing two different evaluations of a modified (`{a} la Borcherds) higher Siegel theta lift on even lattices of signature $(r,s)$, we prove Eichler--Selberg type relations for a wide class of negative weight vector-valued mock modular forms. In doing so, we detail several properties of the lift, as well as showing that it produces an infinite family of local (and locally harmonic) Maa{ss} forms on Grassmanians in certain signatures.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47011639","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}
Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Using different techniques, we improve on this result by constructing factorially-many (again in $g$) such orbits. These new orbits are chosen so that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami. We collect some rudimentary experimental data on the corresponding $SL(2, mathbb{Z})$-orbits and suggest further study and conjectures.
{"title":"Origamis associated to minimally intersecting filling pairs","authors":"Tarik Aougab, W. Menasco, M. Nieland","doi":"10.2140/pjm.2022.317.1","DOIUrl":"https://doi.org/10.2140/pjm.2022.317.1","url":null,"abstract":"Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Using different techniques, we improve on this result by constructing factorially-many (again in $g$) such orbits. These new orbits are chosen so that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami. We collect some rudimentary experimental data on the corresponding $SL(2, mathbb{Z})$-orbits and suggest further study and conjectures.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49573527","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 : 2021-08-08DOI: 10.2140/pjm.2022.318.339
Dihua Jiang, Zhilin Luo
. For a split reductive group G over a number field k , let ρ be an n -dimensional complex representation of its complex dual group G ∨ ( C ). For any irreducible cuspidal automorphic representation σ of G ( A ), where A is the ring of adeles of k , in [JL21], the authors introduce the ( σ, ρ )-Schwartz space S σ,ρ ( A × ) and ( σ, ρ )-Fourier operator F σ,ρ , and study the ( σ, ρ, ψ )-Poisson summation formula on GL 1 , under the assumption that the local Langlands functoriality holds for the pair ( G, ρ ) at all local places of k , where ψ is a non-trivial additive character of k A . Such general formulae on GL 1 , as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture ([ L70]) on global functional equation for the automorphic L -functions L ( s, σ, ρ ). In order to understand such Poisson summation formulae, we continue with [JL21] and develop a further local theory related to the ( σ, ρ )-Schwartz space S σ,ρ ( A × ) and ( σ, ρ )-Fourier operator F σ,ρ . More precisely, over any local field k ν of k , we define distribution kernel functions k σ ν ,ρ,ψ ν ( x ) on GL 1 that represent the ( σ ν , ρ )-Fourier operators F σ ν ,ρ,ψ ν as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands γ -functions γ ( s, σ ν , ρ, ψ ν ) as Mellin transform of the kernel functions. As a consequence, we show that any local Langlands γ -functions are the gamma functions in the sense of I. and I. Piatetski-Shapiro in [GGPS] and of A. Weil in [W66].
{"title":"Certain Fourier operators on GL1 and local\u0000Langlands gamma functions","authors":"Dihua Jiang, Zhilin Luo","doi":"10.2140/pjm.2022.318.339","DOIUrl":"https://doi.org/10.2140/pjm.2022.318.339","url":null,"abstract":". For a split reductive group G over a number field k , let ρ be an n -dimensional complex representation of its complex dual group G ∨ ( C ). For any irreducible cuspidal automorphic representation σ of G ( A ), where A is the ring of adeles of k , in [JL21], the authors introduce the ( σ, ρ )-Schwartz space S σ,ρ ( A × ) and ( σ, ρ )-Fourier operator F σ,ρ , and study the ( σ, ρ, ψ )-Poisson summation formula on GL 1 , under the assumption that the local Langlands functoriality holds for the pair ( G, ρ ) at all local places of k , where ψ is a non-trivial additive character of k A . Such general formulae on GL 1 , as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture ([ L70]) on global functional equation for the automorphic L -functions L ( s, σ, ρ ). In order to understand such Poisson summation formulae, we continue with [JL21] and develop a further local theory related to the ( σ, ρ )-Schwartz space S σ,ρ ( A × ) and ( σ, ρ )-Fourier operator F σ,ρ . More precisely, over any local field k ν of k , we define distribution kernel functions k σ ν ,ρ,ψ ν ( x ) on GL 1 that represent the ( σ ν , ρ )-Fourier operators F σ ν ,ρ,ψ ν as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands γ -functions γ ( s, σ ν , ρ, ψ ν ) as Mellin transform of the kernel functions. As a consequence, we show that any local Langlands γ -functions are the gamma functions in the sense of I. and I. Piatetski-Shapiro in [GGPS] and of A. Weil in [W66].","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42077823","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 : 2021-08-04DOI: 10.2140/pjm.2022.320.177
Ian Wagner
. We define a new class of functions, connected to the classical Laguerre-P´olya class, which we call the shifted Laguerre-P´olya class. Recent work of Griffin, Ono, Rolen, and Zagier shows that the Riemann Xi function is in this class. We prove that a function being in this class is equivalent to its Taylor coefficients, once shifted, being a degree d multiplier sequence for every d , which is equivalent to its shifted coefficients satisfying all of the higher Tur´an inequalities. This mirrors a classical result of P´olya and Schur. For each function in this class we show some order derivative satisfies each extended Laguerre inequality. Finally, we discuss some old and new conjectures about iterated inequalities for functions in this class.
{"title":"On a new class of Laguerre–Pólya type\u0000functions with applications in number theory","authors":"Ian Wagner","doi":"10.2140/pjm.2022.320.177","DOIUrl":"https://doi.org/10.2140/pjm.2022.320.177","url":null,"abstract":". We define a new class of functions, connected to the classical Laguerre-P´olya class, which we call the shifted Laguerre-P´olya class. Recent work of Griffin, Ono, Rolen, and Zagier shows that the Riemann Xi function is in this class. We prove that a function being in this class is equivalent to its Taylor coefficients, once shifted, being a degree d multiplier sequence for every d , which is equivalent to its shifted coefficients satisfying all of the higher Tur´an inequalities. This mirrors a classical result of P´olya and Schur. For each function in this class we show some order derivative satisfies each extended Laguerre inequality. Finally, we discuss some old and new conjectures about iterated inequalities for functions in this class.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48062990","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}
The algebraicity of critical values of triple product $L$-functions in the balanced case was proved by Garrett and Harris, under the assumption that the critical points are on the right and away from center of the critical strip. The missing right-half critical points correspond to certain holomorphic Eisenstein series outside the range of absolute convergence. The remaining difficulties are construction of these holomorphic Eisenstein series and verification of the non-vanishing of the corresponding non-archimedean local zeta integrals. In this paper, we address these problems and complement the result of Garrett and Harris to all critical points. As a consequence, we obtain new cases of Deligne's conjecture for symmetric cube $L$-functions of Hilbert modular forms.
{"title":"Algebraicity of critical values of triple product\u0000L-functions in the balanced case","authors":"Shih-Yu Chen","doi":"10.2140/pjm.2022.321.73","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.73","url":null,"abstract":"The algebraicity of critical values of triple product $L$-functions in the balanced case was proved by Garrett and Harris, under the assumption that the critical points are on the right and away from center of the critical strip. The missing right-half critical points correspond to certain holomorphic Eisenstein series outside the range of absolute convergence. The remaining difficulties are construction of these holomorphic Eisenstein series and verification of the non-vanishing of the corresponding non-archimedean local zeta integrals. In this paper, we address these problems and complement the result of Garrett and Harris to all critical points. As a consequence, we obtain new cases of Deligne's conjecture for symmetric cube $L$-functions of Hilbert modular forms.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47202160","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}
{"title":"Existence and uniqueness of optimal transport maps obtained by the secondary variational method","authors":"Ping Chen, Hairong Liu, Xiao-Ping Yang","doi":"10.2140/pjm.2021.312.75","DOIUrl":"https://doi.org/10.2140/pjm.2021.312.75","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41370832","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 : 2021-08-04DOI: 10.2140/pjm.2021.312.103
Huabin Ge, B. Hua, Wenfeng Jiang
{"title":"The limit of first eigenfunctions of the\u0000p-Laplacian on graphs","authors":"Huabin Ge, B. Hua, Wenfeng Jiang","doi":"10.2140/pjm.2021.312.103","DOIUrl":"https://doi.org/10.2140/pjm.2021.312.103","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42238988","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 : 2021-07-31DOI: 10.2140/pjm.2023.322.139
Mark Reeder
Let $mathfrak g$ be a simple complex Lie algebra of finite dimension. This paper gives an inequality relating the order of an automorphism of $mathfrak g$ to the dimension of its fixed-point subalgebra, and characterizes those automorphisms of $mathfrak g$ for which equality occurs. This is amounts to an inequality/equality for Thomae's function on the group of automorphisms of $mathfrak g$. The result has applications to characters of zero weight spaces, graded Lie algebras, and inequalities for adjoint Swan conductors.
{"title":"Thomae’s function on a Lie group","authors":"Mark Reeder","doi":"10.2140/pjm.2023.322.139","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.139","url":null,"abstract":"Let $mathfrak g$ be a simple complex Lie algebra of finite dimension. This paper gives an inequality relating the order of an automorphism of $mathfrak g$ to the dimension of its fixed-point subalgebra, and characterizes those automorphisms of $mathfrak g$ for which equality occurs. This is amounts to an inequality/equality for Thomae's function on the group of automorphisms of $mathfrak g$. The result has applications to characters of zero weight spaces, graded Lie algebras, and inequalities for adjoint Swan conductors.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41959524","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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