Taiwanese Journal of Mathematics

$L$-series for Vector-valued Modular Forms

Byungchan Kim and Subong Lim

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Motivated by the recent works of Bringmann, Guerzhoy, Kent, and Ono [4] and Bringmann, Fricke, and Kent [3], we introduce $L$-series for vector-valued weakly holomorphic cusp forms, and mock modular period polynomials for vector-valued harmonic weak Maass forms. In particular, we will discuss an integral representation of this new $L$-series and the limiting behavior of special values. Moreover, we also give relations between mock modular periods and $L$-series for vector-valued harmonic weak Maass forms.

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Taiwanese J. Math., Volume 20, Number 4 (2016), 705-722.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

$L$-series vector-valued modular forms vector-valued harmonic weak Maass forms period polynomials


Kim, Byungchan; Lim, Subong. $L$-series for Vector-valued Modular Forms. Taiwanese J. Math. 20 (2016), no. 4, 705--722. doi:10.11650/tjm.20.2016.5976. https://projecteuclid.org/euclid.twjm/1498874486

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  • G. Bol, Invarianten linearer Differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), nos. 3-4, 1–28.
  • R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562.
  • K. Bringmann, K.-H. Fricke and Z. A. Kent, Special $L$-values and periods of weakly holomorphic modular forms, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3425–3439.
  • K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono, Eichler-Shimura theory for mock modular forms, Math. Ann. 355 (2013), no. 3, 1085–1121.
  • J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90.
  • J. H. Bruinier, K. Ono and R. C. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), no. 3, 673–693.
  • J. H. Bruinier and O. Stein, The Weil representation and Hecke operators for vector valued modular forms, Math. Z. 264 (2010), no. 2, 249–270.
  • M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), no. 1, 267–298.
  • M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics 55, Birkhäuser Boston, Boston, 1985.
  • S. Jin and S. Lim, On sign changes of Jacobi forms, preprint.
  • S. Jin, J. Lim and S. Lim, Hecke bound and cuspidality of vector-valued modular forms, preprint.
  • M. Knopp and G. Mason, On vector-valued modular forms and their Fourier coefficients, Acta Arith. 110 (2003), no. 2, 117–124.