Abstract
We give an explicit formula for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin–Selberg series at integers of half-integral weight modular forms, which was not treated by Shimura (J Math Soc Japan 33:649–672, 1981).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berndt, B.C., Evans, R.J.: Sums of Gauss, Jacobi, and Jacobsthal. J. Number Theory 11, 349–398 (1979)
Böcherer, S.: Bemerkungen über die Dirichletreichen von Koecher und Maaß, Math. Gottingensis des Schrift. des SFB. Geometry and Analysis Heft 68 (1986)
Böcherer, S., Schulze-Pillot, R.: The Dirichlet series of Koecher–Maaß and modular forms of weight 3/2. Math. Z. 209, 273–287 (1992)
Choie, Y., Kohnen, W.: Special values of Koecher–Maaß series of Siegel cusp forms. Pac. J. Math. 198, 373–383 (2001)
Duke, W., Imamo\(\bar{g}\)lu, Ö.: A converse theorem and the Saito–Kurokawa lift. Int. Math. Res. Notices 7, 347–355 (1996)
Ikeda, T.: On the lifting of elliptic modular forms to Siegel cusp forms of degree 2n. Ann. Math. 154, 641–681 (2001)
Ibukiyama, T., Katsurada, H.: An explicit formula for Koecher–Maaß Dirichlet series for Eisenstein series of Klingen type. J. Number Theory 102, 223–256 (2003)
Ibukiyama, T., Katsurada, H.: An explicit formula for the Koecher–Maass Dirichlet series for the Ikeda lifting. Abh. Math. Sem. Univ. Hamb. 74, 101–121 (2004)
Ibukiyama, T., Katsurada, H.: Koecher–Maaß series for real analytic Siegel Eisenstein series. In: Automorphic Forms and Zeta Functions, Proceedings of the Conference in Memory of Tsuneo Arakawa, pp. 170–197. World Scientific (2006)
Katsurada, H.: On the special values of certain L-series related to half-integral weight modular forms. In: Proceedings in Mathematics and Statistics. Springer (to appear)
Katsurada, H., Mizuno, Y.: Linear dependence of certain L-values of half-integral weight modular forms. J. Lond. Math. 85, 455–471 (2012)
Kitaoka, Y.: Dirichlet series in the theory of Siegel modular forms. Nagoya Math. J. 95, 73–84 (1984)
Kitaoka, Y.: Arithmetic of Quadratic Forms, Cambridge Tracts in Mathematics, vol. 106. Cambridge University Press, Cambridge (1993)
Kohnen, W.: Modular forms of half-integral weight on \(\varGamma _0(4)\). Math. Ann. 48, 249–266 (1980)
Saito, H.: On \(L\)-functions associated with the vector space of binary quadratic forms. Nagoya Math. J. 130, 149–176 (1993)
Saito, H.: Explicit formula of orbital \(p\)-adic Zeta functions associated to symmetric and Hermitian matrices. Comment. Math. Univ. St. Paul 46, 175–216 (1997)
Shimura, G.: The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math. 29, 65–76 (1976)
Shimura, G.: On the periods of modular forms. Math. Ann. 229, 211–221 (1977)
Shimura, G.: The critical values of certain zeta functions associated with modular forms of half-integral weight. J. Math. Soc. Japan 33, 649–672 (1981)
Acknowledgments
The author thanks the referee for giving useful comments especially on Theorem 5.5 and Proposition 5.10, which make our paper consice.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partly supported by JSPS KAKENHI Grant Number 24540005, JSPS.
Rights and permissions
About this article
Cite this article
Katsurada, H. Explicit formulas for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift and their applications. Math. Z. 276, 1049–1075 (2014). https://doi.org/10.1007/s00209-013-1232-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-013-1232-z
Keywords
- Koecher–Maaß series
- Duke–Imamoglu–Ikeda lift
- Rankin–Selberg series of half-integral weight modular forms