Abstract We define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.
{"title":"ON A BERNSTEIN–SATO POLYNOMIAL OF A MEROMORPHIC FUNCTION","authors":"K. Takeuchi","doi":"10.1017/nmj.2023.10","DOIUrl":"https://doi.org/10.1017/nmj.2023.10","url":null,"abstract":"Abstract We define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48700619","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 Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.
{"title":"ON AN AVERAGE GOLDBACH REPRESENTATION FORMULA OF FUJII","authors":"D. Goldston, A. I. Suriajaya","doi":"10.1017/nmj.2022.44","DOIUrl":"https://doi.org/10.1017/nmj.2022.44","url":null,"abstract":"Abstract Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41532803","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 In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$ . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $ell $ -adic cohomology groups over $overline {{mathbb Q}_{ell }}$ of these Deligne–Lusztig varieties, where $ell not = p$ . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.
{"title":"COHOMOLOGY OF THE BRUHAT–TITS STRATA IN THE UNRAMIFIED UNITARY RAPOPORT–ZINK SPACE OF SIGNATURE \u0000$(1,n-1)$","authors":"Joseph Muller","doi":"10.1017/nmj.2022.39","DOIUrl":"https://doi.org/10.1017/nmj.2022.39","url":null,"abstract":"Abstract In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature \u0000$(1,n-1)$\u0000 . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the \u0000$ell $\u0000 -adic cohomology groups over \u0000$overline {{mathbb Q}_{ell }}$\u0000 of these Deligne–Lusztig varieties, where \u0000$ell not = p$\u0000 . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43740382","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}
This is a contribution to the study of $mathrm {Irr}(G)$ as an $mathrm {Aut}(G)$ -set for G a finite quasisimple group. Focusing on the last open case of groups of Lie type $mathrm {D}$ and $^2mathrm {D}$ , a crucial property is the so-called $A'(infty )$ condition expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in $mathrm {Irr}(G)$ . This is part of the stronger $A(infty )$ condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition $A(infty )$ for groups of type $mathrm {D}$ would still satisfy $A'(infty )$ . This will be used in a second paper to fully establish $A(infty )$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of $G=mathrm {D}_{ l,mathrm {sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.
{"title":"EXTENSIONS OF CHARACTERS IN TYPE D AND THE INDUCTIVE MCKAY CONDITION, I","authors":"Britta Späth","doi":"10.1017/nmj.2023.14","DOIUrl":"https://doi.org/10.1017/nmj.2023.14","url":null,"abstract":"\u0000 This is a contribution to the study of \u0000 \u0000 \u0000 \u0000$mathrm {Irr}(G)$\u0000\u0000 \u0000 as an \u0000 \u0000 \u0000 \u0000$mathrm {Aut}(G)$\u0000\u0000 \u0000 -set for G a finite quasisimple group. Focusing on the last open case of groups of Lie type \u0000 \u0000 \u0000 \u0000$mathrm {D}$\u0000\u0000 \u0000 and \u0000 \u0000 \u0000 \u0000$^2mathrm {D}$\u0000\u0000 \u0000 , a crucial property is the so-called \u0000 \u0000 \u0000 \u0000$A'(infty )$\u0000\u0000 \u0000 condition expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in \u0000 \u0000 \u0000 \u0000$mathrm {Irr}(G)$\u0000\u0000 \u0000 . This is part of the stronger \u0000 \u0000 \u0000 \u0000$A(infty )$\u0000\u0000 \u0000 condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition \u0000 \u0000 \u0000 \u0000$A(infty )$\u0000\u0000 \u0000 for groups of type \u0000 \u0000 \u0000 \u0000$mathrm {D}$\u0000\u0000 \u0000 would still satisfy \u0000 \u0000 \u0000 \u0000$A'(infty )$\u0000\u0000 \u0000 . This will be used in a second paper to fully establish \u0000 \u0000 \u0000 \u0000$A(infty )$\u0000\u0000 \u0000 for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of \u0000 \u0000 \u0000 \u0000$G=mathrm {D}_{ l,mathrm {sc}}(q)$\u0000\u0000 \u0000 extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47130810","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 study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.
{"title":"TRIPLE COVERS OF K3 SURFACES","authors":"Alice Garbagnati, M. Penegini","doi":"10.1017/nmj.2022.15","DOIUrl":"https://doi.org/10.1017/nmj.2022.15","url":null,"abstract":"Abstract We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48659634","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 In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ of the loop algebra $widehat {mathfrak {gl}}_{m|n}$ of ${mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${widehat {{mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ is to determine the action of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $widehat {mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $;{mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ and ${widehat {{mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $mathbb Z$ -form for ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.
{"title":"A REALIZATION OF THE ENVELOPING SUPERALGEBRA \u0000$ {mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$","authors":"J. Du, Qiang Fu, Yanan Lin","doi":"10.1017/nmj.2021.11","DOIUrl":"https://doi.org/10.1017/nmj.2021.11","url":null,"abstract":"Abstract In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum \u0000$mathfrak {gl}_n$\u0000 via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra \u0000${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 of the loop algebra \u0000$widehat {mathfrak {gl}}_{m|n}$\u0000 of \u0000${mathfrak {gl}}_{m|n}$\u0000 with those of affine symmetric groups \u0000${widehat {{mathfrak S}}_{r}}$\u0000 . Then, we give a BLM type realization of \u0000${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 via affine Schur superalgebras. The first application of the realization of \u0000${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 is to determine the action of \u0000${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 on tensor spaces of the natural representation of \u0000$widehat {mathfrak {gl}}_{m|n}$\u0000 . These results in epimorphisms from \u0000$;{mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 to affine Schur superalgebras so that the bridging relation between representations of \u0000${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 and \u0000${widehat {{mathfrak S}}_{r}}$\u0000 is established. As a second application, we construct a Kostant type \u0000$mathbb Z$\u0000 -form for \u0000${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$\u0000 whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48552706","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 Lusztig’s algorithm of computing generalized Green functions of reductive groups involves an ambiguity on certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.
{"title":"GENERALIZED GREEN FUNCTIONS AND UNIPOTENT CLASSES FOR FINITE REDUCTIVE GROUPS, III","authors":"T. Shoji","doi":"10.1017/nmj.2021.12","DOIUrl":"https://doi.org/10.1017/nmj.2021.12","url":null,"abstract":"Abstract Lusztig’s algorithm of computing generalized Green functions of reductive groups involves an ambiguity on certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43444061","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}
Pub Date : 2021-09-01DOI: 10.1017/s0027763020000252
{"title":"NMJ volume 243 Cover and Front matter","authors":"","doi":"10.1017/s0027763020000252","DOIUrl":"https://doi.org/10.1017/s0027763020000252","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46110219","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}
Pub Date : 2021-09-01DOI: 10.1017/s0027763020000264
{"title":"NMJ volume 243 Cover and Back matter","authors":"","doi":"10.1017/s0027763020000264","DOIUrl":"https://doi.org/10.1017/s0027763020000264","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42596078","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 This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $pneq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_varepsilon |$ in the domain away from the singularities when $varepsilon to 0$ , where $u_varepsilon $ is a minimizer of p-GL functional with $p in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $mathbb {R}^2$ .
本文研究了p-Ginzburg-Landau (p-GL)型模型的$pneq 2$。首先，通过奇异性分析得到了整体能量估计和能量集中特性。接下来，我们给出了在远离奇异点的区域中$1-|u_varepsilon |$的衰减率，当$varepsilon to 0$时，其中$u_varepsilon $是与$p in (1,2)$的p-GL函数的最小值。最后，我们在$mathbb {R}^2$上得到了p-GL方程有限能量解的一个Liouville定理。
{"title":"ENERGY CONCENTRATION PROPERTIES OF A p-GINZBURG–LANDAU MODEL","authors":"Y. Lei","doi":"10.1017/nmj.2021.10","DOIUrl":"https://doi.org/10.1017/nmj.2021.10","url":null,"abstract":"Abstract This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with \u0000$pneq 2$\u0000 . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of \u0000$1-|u_varepsilon |$\u0000 in the domain away from the singularities when \u0000$varepsilon to 0$\u0000 , where \u0000$u_varepsilon $\u0000 is a minimizer of p-GL functional with \u0000$p in (1,2)$\u0000 . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on \u0000$mathbb {R}^2$\u0000 .","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46623309","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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