In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $log Vert f Vert $ of a ${mathrm {PSL}}(2,{mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $log Vert f Vert $ and extended the result to compute a regularized inner product of $log Vert f Vert $ with what amounts to powers of the Hauptmodul of $mathrm {PSL}(2,{mathbb {Z}})$ . In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $Gamma $ is the full modular group ${mathrm {PSL}}(2,{mathbb {Z}})$ , thus reproving the theorems from [2]; and second when $Gamma $ is an Atkin–Lehner group $Gamma _{0}(N)^+$ , where explicit computations of inner products are given for certain levels N when the quotient space $overline {Gamma _{0}(N)^+}backslash mathbb {H}$ has genus zero, one, and two.
{"title":"KRONECKER LIMIT FUNCTIONS AND AN EXTENSION OF THE ROHRLICH–JENSEN FORMULA","authors":"J. Cogdell, J. Jorgenson, L. Smajlovic","doi":"10.1017/nmj.2023.7","DOIUrl":"https://doi.org/10.1017/nmj.2023.7","url":null,"abstract":"\u0000 In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm \u0000 \u0000 \u0000 \u0000$log Vert f Vert $\u0000\u0000 \u0000 of a \u0000 \u0000 \u0000 \u0000${mathrm {PSL}}(2,{mathbb {Z}})$\u0000\u0000 \u0000 modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of \u0000 \u0000 \u0000 \u0000$log Vert f Vert $\u0000\u0000 \u0000 and extended the result to compute a regularized inner product of \u0000 \u0000 \u0000 \u0000$log Vert f Vert $\u0000\u0000 \u0000 with what amounts to powers of the Hauptmodul of \u0000 \u0000 \u0000 \u0000$mathrm {PSL}(2,{mathbb {Z}})$\u0000\u0000 \u0000 . In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group \u0000 \u0000 \u0000 \u0000$Gamma $\u0000\u0000 \u0000 of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when \u0000 \u0000 \u0000 \u0000$Gamma $\u0000\u0000 \u0000 is the full modular group \u0000 \u0000 \u0000 \u0000${mathrm {PSL}}(2,{mathbb {Z}})$\u0000\u0000 \u0000 , thus reproving the theorems from [2]; and second when \u0000 \u0000 \u0000 \u0000$Gamma $\u0000\u0000 \u0000 is an Atkin–Lehner group \u0000 \u0000 \u0000 \u0000$Gamma _{0}(N)^+$\u0000\u0000 \u0000 , where explicit computations of inner products are given for certain levels N when the quotient space \u0000 \u0000 \u0000 \u0000$overline {Gamma _{0}(N)^+}backslash mathbb {H}$\u0000\u0000 \u0000 has genus zero, one, and two.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44403449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Asadollahi, Peter Jørgensen, Sibylle Schroll, H. Treffinger
Abstract Building on the embedding of an n-abelian category $mathscr {M}$ into an abelian category $mathcal {A}$ as an n-cluster-tilting subcategory of $mathcal {A}$ , in this paper, we relate the n-torsion classes of $mathscr {M}$ with the torsion classes of $mathcal {A}$ . Indeed, we show that every n-torsion class in $mathscr {M}$ is given by the intersection of a torsion class in $mathcal {A}$ with $mathscr {M}$ . Moreover, we show that every chain of n-torsion classes in the n-abelian category $mathscr {M}$ induces a Harder–Narasimhan filtration for every object of $mathscr {M}$ . We use the relation between $mathscr {M}$ and $mathcal {A}$ to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in $mathscr {M}$ can be induced by a chain of torsion classes in $mathcal {A}$ . Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.
{"title":"ON HIGHER TORSION CLASSES","authors":"J. Asadollahi, Peter Jørgensen, Sibylle Schroll, H. Treffinger","doi":"10.1017/nmj.2022.8","DOIUrl":"https://doi.org/10.1017/nmj.2022.8","url":null,"abstract":"Abstract Building on the embedding of an n-abelian category \u0000$mathscr {M}$\u0000 into an abelian category \u0000$mathcal {A}$\u0000 as an n-cluster-tilting subcategory of \u0000$mathcal {A}$\u0000 , in this paper, we relate the n-torsion classes of \u0000$mathscr {M}$\u0000 with the torsion classes of \u0000$mathcal {A}$\u0000 . Indeed, we show that every n-torsion class in \u0000$mathscr {M}$\u0000 is given by the intersection of a torsion class in \u0000$mathcal {A}$\u0000 with \u0000$mathscr {M}$\u0000 . Moreover, we show that every chain of n-torsion classes in the n-abelian category \u0000$mathscr {M}$\u0000 induces a Harder–Narasimhan filtration for every object of \u0000$mathscr {M}$\u0000 . We use the relation between \u0000$mathscr {M}$\u0000 and \u0000$mathcal {A}$\u0000 to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in \u0000$mathscr {M}$\u0000 can be induced by a chain of torsion classes in \u0000$mathcal {A}$\u0000 . Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"248 1","pages":"823 - 848"},"PeriodicalIF":0.8,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48564767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $operatorname {Sing}(R)$ , and that the support of an object in $operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.
{"title":"HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS","authors":"C. Negron, J. Pevtsova","doi":"10.1017/nmj.2021.18","DOIUrl":"https://doi.org/10.1017/nmj.2021.18","url":null,"abstract":"Abstract We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category \u0000$operatorname {Sing}(R)$\u0000 , and that the support of an object in \u0000$operatorname {Sing}(R)$\u0000 vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"247 1","pages":"731 - 750"},"PeriodicalIF":0.8,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47414506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tomohiro Okuma, M. Rossi, Kei-ichi Watanabe, KEN-ICHI Yoshida
Abstract Let $(A,mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $mathfrak m$ -primary ideals of A. Unlike $p_g$ -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.
{"title":"NORMAL HILBERT COEFFICIENTS AND ELLIPTIC IDEALS IN NORMAL TWO-DIMENSIONAL SINGULARITIES","authors":"Tomohiro Okuma, M. Rossi, Kei-ichi Watanabe, KEN-ICHI Yoshida","doi":"10.1017/nmj.2022.5","DOIUrl":"https://doi.org/10.1017/nmj.2022.5","url":null,"abstract":"Abstract Let \u0000$(A,mathfrak m)$\u0000 be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed \u0000$mathfrak m$\u0000 -primary ideals of A. Unlike \u0000$p_g$\u0000 -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"248 1","pages":"779 - 800"},"PeriodicalIF":0.8,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46960678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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