Special values of shifted Dirichlet series and quasi-modular forms

Wang, Wei

doi:10.1090/proc/17009

Special values of shifted Dirichlet series and quasi-modular forms
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by Wei Wang;

Proc. Amer. Math. Soc. 152 (2024), 4633-4644

DOI: https://doi.org/10.1090/proc/17009

Published electronically: September 24, 2024

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Abstract:

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form.

References

Mrityunjoy Charan, On the adjoint of higher order Serre derivatives, Res. Number Theory 9 (2023), no. 2, Paper No. 30, 8. MR 4576980, DOI 10.1007/s40993-023-00438-w
YoungJu Choie, Hyun Kwang Kim, and Marvin Knopp, Construction of Jacobi forms, Math. Z. 219 (1995), no. 1, 71–76. MR 1340849, DOI 10.1007/BF02572350
Sebastián Daniel Herrero, The adjoint of some linear maps constructed with the Rankin-Cohen brackets, Ramanujan J. 36 (2015), no. 3, 529–536. MR 3317870, DOI 10.1007/s11139-013-9536-5
Abhash Kumar Jha and Arvind Kumar, Construction of cusp forms using Rankin-Cohen brackets, Analytic number theory, modular forms and $q$-hypergeometric series, Springer Proc. Math. Stat., vol. 221, Springer, Cham, 2017, pp. 329–341. MR 3773926, DOI 10.1007/978-3-319-68376-8_{1}
Abhash Kumar Jha and Brundaban Sahu, Rankin-Cohen brackets on Siegel modular forms and special values of certain Dirichlet series, Ramanujan J. 44 (2017), no. 1, 63–73. MR 3696134, DOI 10.1007/s11139-016-9783-3
Abhash Kumar Jha and Brundaban Sahu, Differential operators on Jacobi forms and special values of certain Dirichlet series, Automorphic forms and related topics, Contemp. Math., vol. 732, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 91–99. MR 3973284, DOI 10.1090/conm/732/14793
Masanobu Kaneko and Don Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165–172. MR 1363056, DOI 10.1007/978-1-4612-4264-2_{6}
Winfried Kohnen, Cusp forms and special values of certain Dirichlet series, Math. Z. 207 (1991), no. 4, 657–660. MR 1119963, DOI 10.1007/BF02571414
Arvind Kumar, The adjoint map of the Serre derivative and special values of shifted Dirichlet series, J. Number Theory 177 (2017), 516–527. MR 3629254, DOI 10.1016/j.jnt.2017.01.011
Moni Kumari and Brundaban Sahu, Rankin-Cohen brackets on Hilbert modular forms and special values of certain Dirichlet series, Funct. Approx. Comment. Math. 58 (2018), no. 2, 257–268. MR 3816079, DOI 10.7169/facm/1703
Min Ho Lee, Siegel cusp forms and special values of Dirichlet series of Rankin type, Complex Variables Theory Appl. 31 (1996), no. 2, 97–103. MR 1423243, DOI 10.1080/17476939608814951
Min Ho Lee, Hilbert cusp forms and special values of Dirichlet series of Rankin type, Glasgow Math. J. 40 (1998), no. 1, 71–77. MR 1612165, DOI 10.1017/S0017089500032365
Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103. MR 2409678, DOI 10.1007/978-3-540-74119-0_{1}