Pub Date : 2021-10-18DOI: 10.1017/S1474748021000499
A. Nair, Ankit Rai
Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math. 192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math. 125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.
摘要我们证明了对称空间的非紧同余商的上同调的Oda型限制映射的内射性。这包括了（1）同余实双曲流形、（2）同余复双曲流形和（3）正交Shimura变种之间的约束结果。这些结果推广了Bergeron和Clozel[Quelques conséSequences des travaux d’Arthur pour le spectre et la topologie des variétés双曲线，Invent.Math.192（2013），505–532]和Venkataramana[紧致局部对称空间的同调，Compos.Math.125（2001），221–253]关于紧致同调商的结果。这些证明结合了混合Hodge理论的技术和涉及自同构形式的方法。
{"title":"AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS","authors":"A. Nair, Ankit Rai","doi":"10.1017/S1474748021000499","DOIUrl":"https://doi.org/10.1017/S1474748021000499","url":null,"abstract":"Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math. 192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math. 125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"22 1","pages":"1655 - 1702"},"PeriodicalIF":0.9,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48229281","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 : 2021-09-29DOI: 10.1017/S1474748022000184
Andrei Neguț
Abstract We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.
{"title":"SHUFFLE ALGEBRAS FOR QUIVERS AND R-MATRICES","authors":"Andrei Neguț","doi":"10.1017/S1474748022000184","DOIUrl":"https://doi.org/10.1017/S1474748022000184","url":null,"abstract":"Abstract We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"22 1","pages":"2583 - 2618"},"PeriodicalIF":0.9,"publicationDate":"2021-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41313744","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 : 2021-09-17DOI: 10.1017/s147474802200038x
Frederik Broucke, Gregory Debruyne, J. Vindas
We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $alpha in (0,1]$ and $c>0$ (with $cleq 1$ if $alpha =1$ ), a generalized number system is constructed with Riemann prime counting function $ Pi (x)= operatorname {mathrm {Li}}(x)+ O(xexp (-c log ^{alpha } x ) +log _{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=rho x + Omega _{pm }(xexp (- c'(log xlog _{2} x)^{frac {alpha }{alpha +1}})$ for any $c'>(c(alpha +1))^{frac {1}{alpha +1}}$ , where $rho>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].
{"title":"THE OPTIMAL MALLIAVIN-TYPE REMAINDER FOR BEURLING GENERALIZED INTEGERS","authors":"Frederik Broucke, Gregory Debruyne, J. Vindas","doi":"10.1017/s147474802200038x","DOIUrl":"https://doi.org/10.1017/s147474802200038x","url":null,"abstract":"\u0000\t <jats:p>We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$alpha in (0,1]$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$c>0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> (with <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$cleq 1$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> if <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$alpha =1$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>), a generalized number system is constructed with Riemann prime counting function <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$ Pi (x)= operatorname {mathrm {Li}}(x)+ O(xexp (-c log ^{alpha } x ) +log _{2}x), $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and whose integer counting function satisfies the extremal oscillation estimate <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$N(x)=rho x + Omega _{pm }(xexp (- c'(log xlog _{2} x)^{frac {alpha }{alpha +1}})$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> for any <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$c'>(c(alpha +1))^{frac {1}{alpha +1}}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, where <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802200038X_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$rho>0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].</jats:p>","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41531194","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 : 2021-09-14DOI: 10.1017/S1474748022000482
U. Derenthal, Florian Wilsch
In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $mathbf {A}_1+mathbf {A}_3$ and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.
{"title":"INTEGRAL POINTS ON SINGULAR DEL PEZZO SURFACES","authors":"U. Derenthal, Florian Wilsch","doi":"10.1017/S1474748022000482","DOIUrl":"https://doi.org/10.1017/S1474748022000482","url":null,"abstract":"\u0000 In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type \u0000 \u0000 \u0000 \u0000$mathbf {A}_1+mathbf {A}_3$\u0000\u0000 \u0000 and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47987924","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/s1474748020000468
{"title":"JMJ volume 20 Issue 5 Cover and Front matter","authors":"","doi":"10.1017/s1474748020000468","DOIUrl":"https://doi.org/10.1017/s1474748020000468","url":null,"abstract":"","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"20 1","pages":"f1 - f2"},"PeriodicalIF":0.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41876710","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/s147474802000047x
{"title":"JMJ volume 20 Issue 5 Cover and Back matter","authors":"","doi":"10.1017/s147474802000047x","DOIUrl":"https://doi.org/10.1017/s147474802000047x","url":null,"abstract":"","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":" ","pages":"b1 - b2"},"PeriodicalIF":0.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45878121","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 : 2021-07-28DOI: 10.1017/S1474748022000111
Tudor Pădurariu
Abstract Given a quiver with potential $(Q,W)$ , Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of $(Q,W)$ . Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair $(widetilde{Q}, widetilde{W})$ whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian $Y_{text {MO}}(mathfrak {g}_{Q})$ . For a quiver with potential $(Q,W)$ , we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for $(widetilde{Q}, widetilde{W})$ to recover the positive part of quantum affine algebra $U_{q}(widehat {mathfrak {g}_{Q}})$ defined by Okounkov–Smirnov.
{"title":"CATEGORICAL AND K-THEORETIC HALL ALGEBRAS FOR QUIVERS WITH POTENTIAL","authors":"Tudor Pădurariu","doi":"10.1017/S1474748022000111","DOIUrl":"https://doi.org/10.1017/S1474748022000111","url":null,"abstract":"Abstract Given a quiver with potential \u0000$(Q,W)$\u0000 , Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of \u0000$(Q,W)$\u0000 . Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair \u0000$(widetilde{Q}, widetilde{W})$\u0000 whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian \u0000$Y_{text {MO}}(mathfrak {g}_{Q})$\u0000 . For a quiver with potential \u0000$(Q,W)$\u0000 , we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for \u0000$(widetilde{Q}, widetilde{W})$\u0000 to recover the positive part of quantum affine algebra \u0000$U_{q}(widehat {mathfrak {g}_{Q}})$\u0000 defined by Okounkov–Smirnov.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"22 1","pages":"2717 - 2747"},"PeriodicalIF":0.9,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45308323","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 : 2021-07-22DOI: 10.1017/S1474748021000268
Bingyong Xie
Abstract We propose a conjecture that the Galois representation attached to every Hilbert modular form is noncritical and prove it under certain conditions. Under the same condition we prove Chida, Mok and Park’s conjecture that Fontaine-Mazur L-invariant and Teitelbaum-type L-invariant coincide with each other.
{"title":"ON NONCRITICAL GALOIS REPRESENTATIONS","authors":"Bingyong Xie","doi":"10.1017/S1474748021000268","DOIUrl":"https://doi.org/10.1017/S1474748021000268","url":null,"abstract":"Abstract We propose a conjecture that the Galois representation attached to every Hilbert modular form is noncritical and prove it under certain conditions. Under the same condition we prove Chida, Mok and Park’s conjecture that Fontaine-Mazur L-invariant and Teitelbaum-type L-invariant coincide with each other.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"22 1","pages":"383 - 420"},"PeriodicalIF":0.9,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S1474748021000268","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43712691","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 : 2021-07-21DOI: 10.1017/s1474748022000391
A. Defant, M. Mastyło, A. Pérez-Hernández
We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $mathbb {T}^n$ or Boolean cubes ${-1,1}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.
{"title":"VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV","authors":"A. Defant, M. Mastyło, A. Pérez-Hernández","doi":"10.1017/s1474748022000391","DOIUrl":"https://doi.org/10.1017/s1474748022000391","url":null,"abstract":"\u0000 We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus \u0000 \u0000 \u0000 \u0000$mathbb {T}^n$\u0000\u0000 \u0000 or Boolean cubes \u0000 \u0000 \u0000 \u0000${-1,1}^N$\u0000\u0000 \u0000 . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent \u0000 \u0000 \u0000 \u0000$ell _1$\u0000\u0000 \u0000 -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42327671","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 : 2021-07-14DOI: 10.1017/s1474748022000317
S. Finashin, V. Kharlamov
We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$ , but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.
{"title":"COMBINED COUNT OF REAL RATIONAL CURVES OF CANONICAL DEGREE 2 ON REAL DEL PEZZO SURFACES WITH","authors":"S. Finashin, V. Kharlamov","doi":"10.1017/s1474748022000317","DOIUrl":"https://doi.org/10.1017/s1474748022000317","url":null,"abstract":"\u0000 We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class \u0000 \u0000 \u0000 \u0000$-2K$\u0000\u0000 \u0000 , but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43410165","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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