Abstract
Hulek and others conjectured that the unique differential three-form F (up to scalar) on the Siegel threefold associated to the group Γ1,3(2) comes from the Saito-Kurokawa lift of the elliptic newform h of weight 4 for Γ0(6). This F have been already constructed as a Borcherds product (cf. Gritsenko and Hulek in Int Math Res Notices 17:915–937, 1999). In this paper, we prove this conjecture by using the Yoshida lift and we settle a conjecture which relates our theorem. A remarkable fact is that the Yoshida lift using the usual test function cannot give the Saito-Kurokawa type lift of weight 3 associated to the group Γ1,3(2). So important task is to find special test functions for the Yoshida lift at the bad primes 2 and 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arthur, J.: Unipotent automorphic representations: conjectures. Asterisque 171–172, 13–71 (1989)
Arthur, J.: Automorphic representations of GSp(4), Contributions to automorphic forms, geometry, and number theory, pp. 65–81. Johns Hopkins University Press, Baltimore (2004)
Böcherer S. and Schulze-Pillot R. (1997). Siegel modular forms and theta series attached to quaternion algebras II. Nagoya Math. J. 147: 71–106
Casselman W. (1973). On some results of Atkin and Lehner. Math. Ann. 201: 301–314
Casselman W. (1972). An assortment of results on representations of GL2(k), in Modular Functions of One Variable II. L.N.M. 349: 1–54
Dokchitser T. (2004). Computing special values of motivic L-functions. Exp. Math. 13(2): 137–149
Gelbart S. (1975). Automorphic forms on adele groups. Annals of Mathematics Studies, no 83. Princeton University Press, Princeton
Gritsenko V. and Hulek K. (1999). The modular form of the Barth–Nieto quintic. Int. Math. Res. Notices 17: 915–937
Hulek K., Spandaw J., Geemen B. and Straten D. (2001). The modularity of the Barth-Nieto quintic and its relatives. Adv. Geometry 1: 263–289
Jacquet, H., Langlands, R.P.: Automorphic forms on GL(2). L.N.M. 114
Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture I. Preprint (downloadable from Khare’s homepage)
Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture II. Preprint (downloadable from Khare’s homepage)
Kisin, M.: Moduli of finite flat group schemes and modularity. Preprint
Okazaki T., Salvati Manni R. and Top’s J. (2006). conjectures on Siegel modular forms and Abelian surfaces. Am. J. Math. 128: 139–165
Okazaki T. (2007). L-functions of S 3(Γ(4, 8)). J. Number Thoery 125: 117–132
Okazaki, T.: Degree 4 spinor L-function of base change lift type. Preprint
Pizer A. (1980). An algorithm for computing modular forms on Γ0(N). J. Algebra 64: 340–380
Nygaard N.O. and Geemen B. (1995). On the geometry and arithmetic of some Siegel modular threefolds. J. Number Theory 53: 45–87
Waldspurger J.L. (1991). Correspondence de Shimura et quaternions. Forum Math. 3: 219–307
Yoshida H. (1980). Siegel’s modular forms and the arithmetics of quadratic forms. Invent. Math. 60: 193–248
Yoshida H. (1984). On Siegel modular forms obtained by theta series. J. Reine. Angrew. Math. 352: 184–219
Yoshida H. (1979). Weil’s representations of the symplectic groups over finite fields. J. Math. Soc. Jpn 31(2): 399–426
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Tomoyoshi Ibukiyama on his 60th birthday.
Rights and permissions
About this article
Cite this article
Okazaki, T., Yamauchi, T. A Siegel modular threefold and Saito-Kurokawa type lift to S 3(Γ1,3(2)). Math. Ann. 341, 589–601 (2008). https://doi.org/10.1007/s00208-007-0204-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0204-1