Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.
{"title":"Cluster algebra structures on Poisson nilpotent algebras","authors":"K. Goodearl, M. Yakimov","doi":"10.1090/memo/1445","DOIUrl":"https://doi.org/10.1090/memo/1445","url":null,"abstract":"Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135707798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this monograph is to study the global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small smooth and mildly decaying at infinity. Some physical models strictly related to general relativity have shown the importance of studying such systems but very few results are known at present in low space dimension. We study here a model two-dimensional system, in which the nonlinearity writes in terms of “null forms”, and show the global existence of small solutions. Our goal is to prove some energy estimates on the solution when a certain number of Klainerman vector fields is acting on it, and some optimal uniform estimates. The former ones are obtained using systematically quasilinear normal forms, in their para-differential version; the latter ones are recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system by means of a semiclassical micro-local analysis of the problem. We expect the strategy developed here to be robust enough to enable us, in the future, to treat the case of the most general nonlinearities.
{"title":"Global Existence of Small Amplitude Solutions for a Model Quadratic Quasilinear Coupled Wave-Klein-Gordon System in Two Space Dimension, with Mildly Decaying Cauchy Data","authors":"Annalaura Stingo","doi":"10.1090/memo/1441","DOIUrl":"https://doi.org/10.1090/memo/1441","url":null,"abstract":"The aim of this monograph is to study the global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small smooth and mildly decaying at infinity. Some physical models strictly related to general relativity have shown the importance of studying such systems but very few results are known at present in low space dimension. We study here a model two-dimensional system, in which the nonlinearity writes in terms of “null forms”, and show the global existence of small solutions. Our goal is to prove some energy estimates on the solution when a certain number of Klainerman vector fields is acting on it, and some optimal uniform estimates. The former ones are obtained using systematically quasilinear normal forms, in their para-differential version; the latter ones are recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system by means of a semiclassical micro-local analysis of the problem. We expect the strategy developed here to be robust enough to enable us, in the future, to treat the case of the most general nonlinearities.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Equipped with the L2,qL^{2,q}-distortion distance DD _{2,q}, the space XX _{2q} of all metric measure spaces (X,d ,m ) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on ol XX _{2q} are presented.
利用l2,q L^{2,q} -畸变距离DD _{2,q}，证明了所有度量测量空间(X，d，m)的空间XX _{2q}在Alexandrov意义上具有非负曲率。详细描述了测地线和切线空间。此外，还给出了半凸泛函的类及其在ol XX _{2q}上的梯度流。
{"title":"The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces","authors":"Karl-Theodor Sturm","doi":"10.1090/memo/1443","DOIUrl":"https://doi.org/10.1090/memo/1443","url":null,"abstract":"Equipped with the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 2 comma q\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^{2,q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-distortion distance <inline-formula content-type=\"math/tex\"> <tex-math> DD _{2,q}</tex-math></inline-formula>, the space <inline-formula content-type=\"math/tex\"> <tex-math> XX _{2q}</tex-math></inline-formula> of all metric measure spaces <inline-formula content-type=\"math/tex\"> <tex-math> (X,d ,m )</tex-math></inline-formula> is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on <inline-formula content-type=\"math/tex\"> <tex-math> ol XX _{2q}</tex-math></inline-formula> are presented.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The primary goal of this work is to construct p p -adic families of modular forms of half-integral weight, by using Waldspurger’s automorphic framework to make the results as comprehensive and precise as possible. A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight. As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani’s lift vanishes. We also give a number of results on test vectors for ramified representations which are of independent interest.
{"title":"Toric Periods and 𝑝-adic Families of Modular Forms of Half-Integral Weight","authors":"V. Vatsal","doi":"10.1090/memo/1438","DOIUrl":"https://doi.org/10.1090/memo/1438","url":null,"abstract":"The primary goal of this work is to construct \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic families of modular forms of half-integral weight, by using Waldspurger’s automorphic framework to make the results as comprehensive and precise as possible. A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight. As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani’s lift vanishes. We also give a number of results on test vectors for ramified representations which are of independent interest.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41455776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species. Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures. We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences. We highlight some main applications: We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account. We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of one such result for generalized permutahedra. We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday. We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions. We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus.
{"title":"Hopf Monoids and Generalized Permutahedra","authors":"Marcelo Aguiar, Federico Ardila","doi":"10.1090/memo/1437","DOIUrl":"https://doi.org/10.1090/memo/1437","url":null,"abstract":"Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species. Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures. We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences. We highlight some main applications: We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account. We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of one such result for generalized permutahedra. We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday. We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions. We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134971547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C∞C^{infty }-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations.
We first introduce the notion of hereditary C∞C^infty-paracompactness along with the semiclassicality condition on a C∞C^infty-manifold, which enables us to use local convexity in local arguments. Then, we prove that for C∞C^infty-manifolds MM and NN, the smooth singular complex of the diffeological space <
{"title":"Smooth Homotopy of Infinite-Dimensional 𝐶^{∞}-Manifolds","authors":"Hiroshi Kihara","doi":"10.1090/memo/1436","DOIUrl":"https://doi.org/10.1090/memo/1436","url":null,"abstract":"<p>In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^{infty }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations.</p>\u0000\u0000<p>We first introduce the notion of hereditary <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-paracompactness along with the semiclassicality condition on a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-manifold, which enables us to use local convexity in local arguments. Then, we prove that for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-manifolds <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\u0000 <mml:semantics>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the smooth singular complex of the diffeological space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity Baseline left-parenthesis upper M comma upper N right-parenthesis\">\u0000 <","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44500876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael Hill, Xiaolin Shi, Guozhen Wang, Zhouli Xu
We completely compute the slice spectral sequence of the C4C_4-spectrum BP((C4))⟨2⟩BP^{(!(C_4)!)}langle 2 rangle. This spectrum provides a model for a height-4 Lubin–Tate theory with a C4C_4-action induced from the Goerss–Hopkins–Miller theorem. In particular, our computation shows that E4hC12E_4^{hC_{12}} is 384-periodic.
The finite subgroups of GL4(Z)GL_4(mathbb {Z}) are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist 710710 non-conjugate finite groups in GL4(Z)GL_4(mathbb {Z}). Each finite group GG of GL4(Z)GL_4(mathbb {Z}) acts naturally on
在Brown，Büllow，Neubüser，Wondratscheck和Zassenhaus（1978）中，对G L 4（Z）GL_4（mathbb｛Z｝）的有限子群进行了共轭分类；特别地，在G L4（Z）GL_4（mathbb｛Z｝）中存在710 710个非共轭有限群。G L4（Z）GL_4（mathbb｛Z｝）的每个有限群G G自然作用于ZŞ4mathbb{Z｝^｛oplus 4｝；因此我们得到了一个忠实的G G-格M M，其中r a n k Z M=4{rank}_mathb{Z}M=4。通过这种方式，正好有710 710个这样的晶格。给定一个具有r a n k Z M=4 mathrm的G G-格M M{rank}_mathbb｛Z｝M=4，群G G通过乘法作用作用于有理函数域C（M）≔C（x1，x2，x3，x4）mathbb｛C｝（M） mathbb{C}（x_1，x_2，x_3，x_4），即Cmathbb｛C｝上的纯单体自同构。我们讨论了固定域C（M）Gmathbb｛C｝（M）^G的合理性问题。我们研究的一个工具是域C
{"title":"Multiplicative Invariant Fields of Dimension ≤6","authors":"A. Hoshi, M. Kang, A. Yamasaki","doi":"10.1090/memo/1403","DOIUrl":"https://doi.org/10.1090/memo/1403","url":null,"abstract":"<p>The finite subgroups of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L 4 left-parenthesis double-struck upper Z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GL_4(mathbb {Z})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"710\">\u0000 <mml:semantics>\u0000 <mml:mn>710</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">710</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> non-conjugate finite groups in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L 4 left-parenthesis double-struck upper Z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GL_4(mathbb {Z})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Each finite group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L 4 left-parenthesis double-struck upper Z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GL_4(mathbb {Z})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> acts naturally on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Superscript circled-plus 4\">\u0000 <mml:semantics>\u0000 ","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49579466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Blomer, É. Fouvry, E. Kowalski, P. Michel, Djordje Milićević, W. Sawin
For a fairly general family of L L -functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q q , and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values L ( f ⊗ χ , 1 / 2 ) L(fotimes chi ,1/2) are non-zero, and indeed bounded from below; there exist many characters χ chi for which the central L L -value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.
{"title":"The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions","authors":"V. Blomer, É. Fouvry, E. Kowalski, P. Michel, Djordje Milićević, W. Sawin","doi":"10.1090/memo/1394","DOIUrl":"https://doi.org/10.1090/memo/1394","url":null,"abstract":"For a fairly general family of \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family.\u0000\u0000We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime \u0000\u0000 \u0000 q\u0000 q\u0000 \u0000\u0000, and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values \u0000\u0000 \u0000 \u0000 L\u0000 (\u0000 f\u0000 ⊗\u0000 χ\u0000 ,\u0000 1\u0000 \u0000 /\u0000 \u0000 2\u0000 )\u0000 \u0000 L(fotimes chi ,1/2)\u0000 \u0000\u0000 are non-zero, and indeed bounded from below; there exist many characters \u0000\u0000 \u0000 χ\u0000 chi\u0000 \u0000\u0000 for which the central \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42402079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1975 Figari, Høegh-Krohn and Nappi constructed the P ( φ ) 2 {mathscr P}(varphi )_2 model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs of S O 0 ( 1 , 2 ) SO_0(1,2) for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables. ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations of S O ( 3 ) SO(3) on the Euclidean sphere, and discuss how the P ( φ ) 2 {mathscr P}(varphi )_2 interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physical infrared problems are absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for the P ( φ ) 2 {mathscr P}(varphi )_2 model and we explain how the generators of the boosts and the rotations for the interacting quantum field theory arise from the stress-energy tensor. Finally, we show that the interacting quantum fields satisfy the equations of motion in their covariant form. In summary, we argue that the de Sitter P ( φ ) 2 {mathscr P}(varphi )_2 model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.
{"title":"The 𝒫(𝜑)₂ Model on de Sitter Space","authors":"J. Barata, C. Jäkel, J. Mund","doi":"10.1090/memo/1389","DOIUrl":"https://doi.org/10.1090/memo/1389","url":null,"abstract":"In 1975 Figari, Høegh-Krohn and Nappi constructed the \u0000\u0000 \u0000 \u0000 \u0000 \u0000 P\u0000 \u0000 \u0000 (\u0000 φ\u0000 \u0000 )\u0000 2\u0000 \u0000 \u0000 {mathscr P}(varphi )_2\u0000 \u0000\u0000 model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs of \u0000\u0000 \u0000 \u0000 S\u0000 \u0000 O\u0000 0\u0000 \u0000 (\u0000 1\u0000 ,\u0000 2\u0000 )\u0000 \u0000 SO_0(1,2)\u0000 \u0000\u0000 for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables.\u0000\u0000ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations of \u0000\u0000 \u0000 \u0000 S\u0000 O\u0000 (\u0000 3\u0000 )\u0000 \u0000 SO(3)\u0000 \u0000\u0000 on the Euclidean sphere, and discuss how the \u0000\u0000 \u0000 \u0000 \u0000 \u0000 P\u0000 \u0000 \u0000 (\u0000 φ\u0000 \u0000 )\u0000 2\u0000 \u0000 \u0000 {mathscr P}(varphi )_2\u0000 \u0000\u0000 interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physical infrared problems are absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for the \u0000\u0000 \u0000 \u0000 \u0000 \u0000 P\u0000 \u0000 \u0000 (\u0000 φ\u0000 \u0000 )\u0000 2\u0000 \u0000 \u0000 {mathscr P}(varphi )_2\u0000 \u0000\u0000 model and we explain how the generators of the boosts and the rotations for the interacting quantum field theory arise from the stress-energy tensor. Finally, we show that the interacting quantum fields satisfy the equations of motion in their covariant form. In summary, we argue that the de Sitter \u0000\u0000 \u0000 \u0000 \u0000 \u0000 P\u0000 \u0000 \u0000 (\u0000 φ\u0000 \u0000 )\u0000 2\u0000 \u0000 \u0000 {mathscr P}(varphi )_2\u0000 \u0000\u0000 model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48283879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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