The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product
We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi thet...
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| Published in | Journal of mathematical analysis and applications Vol. 471; no. 1-2; pp. 623 - 646 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
01.03.2019
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| Abstract | We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann–Manschot and about the asymptotic behavior of the χy-genera of Hilbert schemes of points on K3 surfaces by Manschot–Rolon. We also get the algebraicity of the generating functions given by Göttsche for the Hilbert schemes associated to general algebraic surfaces. |
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| AbstractList | We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann–Manschot and about the asymptotic behavior of the χy-genera of Hilbert schemes of points on K3 surfaces by Manschot–Rolon. We also get the algebraicity of the generating functions given by Göttsche for the Hilbert schemes associated to general algebraic surfaces. |
| Author | Jo, Sihun Jin, Seokho |
| Author_xml | – sequence: 1 givenname: Seokho orcidid: 0000-0003-3127-4170 surname: Jin fullname: Jin, Seokho email: archimed@cau.ac.kr organization: Department of Mathematics, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Republic of Korea – sequence: 2 givenname: Sihun surname: Jo fullname: Jo, Sihun email: sihunjo@woosuk.ac.kr organization: Department of Mathematics Education, Woosuk University, 443 Samnye-ro, Samnye-eup, Wanju-Gun, Jeollabuk-do 55338, Republic of Korea |
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| Cites_doi | 10.1007/JHEP02(2016)170 10.1093/qmath/22.1.107 10.4310/CNTP.2013.v7.n3.a4 10.4310/CNTP.2015.v9.n2.a6 10.1017/S0305004100040573 10.1090/tran/6409 10.1007/BF01453572 10.1215/00127094-3449994 |
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| References | Bringmann, Dousse (br0020) 2016; 368 Andrews, Askey, Roy (br0010) 1999; vol. 71 Göttsche (br0040) 1990; 286 Manschot, Rolon (br0090) 2015; 9 Kim, Kim, Lee (br0070) 2016; 2 Macdonald (br0080) 1962; 58 Wright (br0100) 1971; 22 Kubert, Lang (br0060) 1981; vol. 244 Griffin, Ono, Warnaar (br0050) 2016; 165 Bringmann, Manschot (br0030) 2013; 7 Göttsche (10.1016/j.jmaa.2018.10.096_br0040) 1990; 286 Manschot (10.1016/j.jmaa.2018.10.096_br0090) 2015; 9 Bringmann (10.1016/j.jmaa.2018.10.096_br0030) 2013; 7 Griffin (10.1016/j.jmaa.2018.10.096_br0050) 2016; 165 Andrews (10.1016/j.jmaa.2018.10.096_br0010) 1999; vol. 71 Kim (10.1016/j.jmaa.2018.10.096_br0070) 2016; 2 Kubert (10.1016/j.jmaa.2018.10.096_br0060) 1981; vol. 244 Macdonald (10.1016/j.jmaa.2018.10.096_br0080) 1962; 58 Wright (10.1016/j.jmaa.2018.10.096_br0100) 1971; 22 Bringmann (10.1016/j.jmaa.2018.10.096_br0020) 2016; 368 |
| References_xml | – volume: 368 start-page: 3141 year: 2016 end-page: 3155 ident: br0020 article-title: On Dyson's crank conjecture and the uniform asymptotic behavior of certain inverse theta functions publication-title: Trans. Amer. Math. Soc. contributor: fullname: Dousse – volume: vol. 71 year: 1999 ident: br0010 article-title: Special Functions publication-title: Encyclopedia Math. Appl. contributor: fullname: Roy – volume: 58 start-page: 563 year: 1962 end-page: 568 ident: br0080 article-title: The Poincaré polynomial of a symmetric product publication-title: Proc. Camb. Philos. Soc. contributor: fullname: Macdonald – volume: 7 start-page: 497 year: 2013 end-page: 513 ident: br0030 article-title: Asymptotic formulas for coefficients of inverse theta functions publication-title: Commun. Number Theory Phys. contributor: fullname: Manschot – volume: 165 start-page: 1475 year: 2016 end-page: 1527 ident: br0050 article-title: A framework of Rogers–Ramanujan identities and their arithmetic properties publication-title: Duke Math. J. contributor: fullname: Warnaar – volume: vol. 244 year: 1981 ident: br0060 article-title: Modular Units publication-title: Grundlehren Math. Wiss. contributor: fullname: Lang – volume: 9 start-page: 413 year: 2015 end-page: 436 ident: br0090 article-title: The asymptotic profile of publication-title: Commun. Number Theory Phys. contributor: fullname: Rolon – volume: 2 start-page: 170 year: 2016 ident: br0070 article-title: Little strings and T-duality publication-title: J. High Energy Phys. contributor: fullname: Lee – volume: 286 start-page: 193 year: 1990 end-page: 207 ident: br0040 article-title: The Betti numbers of the Hilbert scheme of points on a smooth projective surface publication-title: Math. Ann. contributor: fullname: Göttsche – volume: 22 start-page: 107 year: 1971 end-page: 116 ident: br0100 article-title: Stacks. II publication-title: Q. J. Math. contributor: fullname: Wright – volume: 2 start-page: 170 year: 2016 ident: 10.1016/j.jmaa.2018.10.096_br0070 article-title: Little strings and T-duality publication-title: J. High Energy Phys. doi: 10.1007/JHEP02(2016)170 contributor: fullname: Kim – volume: vol. 71 year: 1999 ident: 10.1016/j.jmaa.2018.10.096_br0010 article-title: Special Functions contributor: fullname: Andrews – volume: 22 start-page: 107 year: 1971 ident: 10.1016/j.jmaa.2018.10.096_br0100 article-title: Stacks. II publication-title: Q. J. Math. doi: 10.1093/qmath/22.1.107 contributor: fullname: Wright – volume: 7 start-page: 497 issue: 3 year: 2013 ident: 10.1016/j.jmaa.2018.10.096_br0030 article-title: Asymptotic formulas for coefficients of inverse theta functions publication-title: Commun. Number Theory Phys. doi: 10.4310/CNTP.2013.v7.n3.a4 contributor: fullname: Bringmann – volume: vol. 244 year: 1981 ident: 10.1016/j.jmaa.2018.10.096_br0060 article-title: Modular Units contributor: fullname: Kubert – volume: 9 start-page: 413 issue: 2 year: 2015 ident: 10.1016/j.jmaa.2018.10.096_br0090 article-title: The asymptotic profile of χy-genera of Hilbert schemes of points on K3 surfaces publication-title: Commun. Number Theory Phys. doi: 10.4310/CNTP.2015.v9.n2.a6 contributor: fullname: Manschot – volume: 58 start-page: 563 year: 1962 ident: 10.1016/j.jmaa.2018.10.096_br0080 article-title: The Poincaré polynomial of a symmetric product publication-title: Proc. Camb. Philos. Soc. doi: 10.1017/S0305004100040573 contributor: fullname: Macdonald – volume: 368 start-page: 3141 issue: 5 year: 2016 ident: 10.1016/j.jmaa.2018.10.096_br0020 article-title: On Dyson's crank conjecture and the uniform asymptotic behavior of certain inverse theta functions publication-title: Trans. Amer. Math. Soc. doi: 10.1090/tran/6409 contributor: fullname: Bringmann – volume: 286 start-page: 193 year: 1990 ident: 10.1016/j.jmaa.2018.10.096_br0040 article-title: The Betti numbers of the Hilbert scheme of points on a smooth projective surface publication-title: Math. Ann. doi: 10.1007/BF01453572 contributor: fullname: Göttsche – volume: 165 start-page: 1475 issue: 8 year: 2016 ident: 10.1016/j.jmaa.2018.10.096_br0050 article-title: A framework of Rogers–Ramanujan identities and their arithmetic properties publication-title: Duke Math. J. doi: 10.1215/00127094-3449994 contributor: fullname: Griffin |
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| Title | The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product |
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