We prove a spectral decomposition formula for averages of Zagier $L$-series in terms of moments of symmetric square $L$-functions associated to Maass and holomorphic cusp forms of levels $4$, $16$, $64$.
{"title":"Spectral decomposition formula and moments of symmetric square $L$-functions","authors":"Olga Germanovna Balkanova","doi":"10.4213/im9330e","DOIUrl":"https://doi.org/10.4213/im9330e","url":null,"abstract":"We prove a spectral decomposition formula for averages of Zagier $L$-series in terms of moments of symmetric square $L$-functions associated to Maass and holomorphic cusp forms of levels $4$, $16$, $64$.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135007564","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 $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $Gamma_{L}^{(v)}$ the ramification subgroups of $Gamma_{L}=operatorname{Gal}(L^{mathrm{sep}}/L)$. We consider the category $operatorname{MGamma}_{L}^{mathrm{Lie}}$ of finite $mathbb{Z}_p[Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $Gamma_L$ in $operatorname{Aut}_{mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $Gamma_L^{(v)}$ in $operatorname{Aut}_{mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $widetilde{Omega} [N]$ on the Fontaine etale $phi $-module $M(H)$ associated with $H$. The forms $widetilde{Omega}[N]$ are completely determined by a canonical connection $nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $mathbb{F}_p[Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $mathbb{G}_m$.
{"title":"Ramification filtration and differential forms","authors":"Viktor Aleksandrovich Abrashkin","doi":"10.4213/im9322e","DOIUrl":"https://doi.org/10.4213/im9322e","url":null,"abstract":"Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $Gamma_{L}^{(v)}$ the ramification subgroups of $Gamma_{L}=operatorname{Gal}(L^{mathrm{sep}}/L)$. We consider the category $operatorname{MGamma}_{L}^{mathrm{Lie}}$ of finite $mathbb{Z}_p[Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $Gamma_L$ in $operatorname{Aut}_{mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $Gamma_L^{(v)}$ in $operatorname{Aut}_{mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $widetilde{Omega} [N]$ on the Fontaine etale $phi $-module $M(H)$ associated with $H$. The forms $widetilde{Omega}[N]$ are completely determined by a canonical connection $nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $mathbb{F}_p[Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a \"good\" lift of a generator of $operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $mathbb{G}_m$.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135440272","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}
Upper moduli part of adjunction is introduced and its basic property are discussed. The moduli part is b-Cartier in the case of rational multiplicities and is b-nef in the maximal case.
{"title":"Log adjunction: moduli part","authors":"Vyacheslav Vladimirovich Shokurov","doi":"10.4213/im9279e","DOIUrl":"https://doi.org/10.4213/im9279e","url":null,"abstract":"Upper moduli part of adjunction is introduced and its basic property are discussed. The moduli part is b-Cartier in the case of rational multiplicities and is b-nef in the maximal case.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135440534","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}
We consider a two-dimensional hierarchical lattice in which the vertices of a square represent an elementary cell. In the generalized hierarchical model, the distance between opposite vertices of a square differs from that between adjacent vertices and is a parameter of the new model. The Gaussian part of the Hamiltonian of the 4-component generalized fermionic hierarchical model is invariant under the block-spin renormalization group transformation. The transformation of the renormalization group in the space of coefficients, which specify the Grassmann-valued density of the free measure, is explicitly calculated as a homogeneous mapping of degree four in the two-dimensional projective space.
{"title":"The renormalization group transformation in the generalized fermionic hierarchical model","authors":"Mukadas Dmukhtasibovich Missarov, Dmitrii Airatovich Khajrullin","doi":"10.4213/im9371e","DOIUrl":"https://doi.org/10.4213/im9371e","url":null,"abstract":"We consider a two-dimensional hierarchical lattice in which the vertices of a square represent an elementary cell. In the generalized hierarchical model, the distance between opposite vertices of a square differs from that between adjacent vertices and is a parameter of the new model. The Gaussian part of the Hamiltonian of the 4-component generalized fermionic hierarchical model is invariant under the block-spin renormalization group transformation. The transformation of the renormalization group in the space of coefficients, which specify the Grassmann-valued density of the free measure, is explicitly calculated as a homogeneous mapping of degree four in the two-dimensional projective space.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661293","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 notion of primitive recursive realizability was introduced by S. Salehi as a kind of semantics for the language of basic arithmetic using primitive recursive functions. It is of interest to study the corresponding predicate logic. D. A. Viter proved that the predicate logic of primitive recursive realizability by Salehi is not arithmetical. The technically complex proof combines the methods used by the author of this article in the study of predicate logics of constructive arithmetic theories and the results of M. Ardeshir on the translation of the intuitionistic predicate logic into the basic predicate logic. The purpose of this article is to present another proof of Viter's result by directly transferring the methods used earlier in proving the nonarithmeticity of the predicate logic of recursive realizability.
原始递归可实现性的概念是由S. Salehi提出的，作为一种基于原始递归函数的基本算术语言的语义。研究相应的谓词逻辑是很有意义的。D. A. Viter用Salehi证明了原始递归可实现性的谓词逻辑不是算术的。技术上复杂的证明结合了本文作者在构造算术理论的谓词逻辑研究中使用的方法和M. Ardeshir关于将直觉谓词逻辑转化为基本谓词逻辑的结果。本文的目的是通过直接转移先前用于证明递归可实现谓词逻辑的非算术性的方法，给出Viter结果的另一种证明。
{"title":"The nonarithmeticity of the predicate logic\u0000of primitive recursive realizability","authors":"V. E. Plisko","doi":"10.4213/im9288e","DOIUrl":"https://doi.org/10.4213/im9288e","url":null,"abstract":"The notion of primitive recursive realizability was introduced by S. Salehi as a kind of semantics for the language of basic arithmetic using primitive recursive functions. It is of interest to study the corresponding predicate logic. D. A. Viter proved that the predicate logic of primitive recursive realizability by Salehi is not arithmetical. The technically complex proof combines the methods used by the author of this article in the study of predicate logics of constructive arithmetic theories and the results of M. Ardeshir on the translation of the intuitionistic predicate logic into the basic predicate logic. The purpose of this article is to present another proof of Viter's result by directly transferring the methods used earlier in proving the nonarithmeticity of the predicate logic of recursive realizability.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70326975","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 paper is a survey devoted to the topological phases - one of the actively developing directions in the theory of solid states. An interpretation of topological phases in terms of the generalized cohomology theories and $K$-theory is given.
{"title":"Topological phases in the theory of solid states","authors":"Armen Glebovich Sergeev, Egor Teplyakov","doi":"10.4213/im9349e","DOIUrl":"https://doi.org/10.4213/im9349e","url":null,"abstract":"The paper is a survey devoted to the topological phases - one of the actively developing directions in the theory of solid states. An interpretation of topological phases in terms of the generalized cohomology theories and $K$-theory is given.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661295","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 case study of fermions and the attempt to deduce their structure from classical probability opens new ways for classical and quantum probability, in particular, for the notion of stochastic coupling which, on the basis of the example of fermions, we enlarge to the notion of algebraic coupling, and for the various notions of stochastic independence. These notions are shown to be strictly correlated with algebraic and stochastic couplings. This approach allows to expand considerably the notion of open system. The above statements will be illustrated with some examples. The last section shows how, from these new stochastic couplings, new statistics emerge alongside the known Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics.
{"title":"Fermions from classical probability and statistics defined by stochastic independence","authors":"Luigi Accardi, Yun Gang Lu","doi":"10.4213/im9389e","DOIUrl":"https://doi.org/10.4213/im9389e","url":null,"abstract":"The case study of fermions and the attempt to deduce their structure from classical probability opens new ways for classical and quantum probability, in particular, for the notion of stochastic coupling which, on the basis of the example of fermions, we enlarge to the notion of algebraic coupling, and for the various notions of stochastic independence. These notions are shown to be strictly correlated with algebraic and stochastic couplings. This approach allows to expand considerably the notion of open system. The above statements will be illustrated with some examples. The last section shows how, from these new stochastic couplings, new statistics emerge alongside the known Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661672","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}
We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.
{"title":"$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory","authors":"Taras Evgenievich Panov, George Chernykh","doi":"10.4213/im9334e","DOIUrl":"https://doi.org/10.4213/im9334e","url":null,"abstract":"We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135007538","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 Michael selection theorem is extended to the case of set-valued mappings with not necessarily convex values. Classical approximation problems on cone-spaces with symmetric and asymmetric seminorms are considered. In particular, conditions for existence of continuous selections for convex subsets of asymmetric spaces are studied. The problem of existence of a Chebyshev centre for a bounded set is solved in a semilinear space consisting of bounded convex sets with Hausdorff semimetric.
{"title":"Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces","authors":"Igor' Germanovich Tsar'kov","doi":"10.4213/im9331e","DOIUrl":"https://doi.org/10.4213/im9331e","url":null,"abstract":"The Michael selection theorem is extended to the case of set-valued mappings with not necessarily convex values. Classical approximation problems on cone-spaces with symmetric and asymmetric seminorms are considered. In particular, conditions for existence of continuous selections for convex subsets of asymmetric spaces are studied. The problem of existence of a Chebyshev centre for a bounded set is solved in a semilinear space consisting of bounded convex sets with Hausdorff semimetric.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135007551","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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