No Access
SIMPLE GRADED RINGS OF SIEGEL MODULAR FORMS, DIFFERENTIAL OPERATORS AND BORCHERDS PRODUCTS
https://doi.org/10.1142/S0129167X05002837Cited by:36
Abstract
In this paper, we show that the graded ring of Siegel modular forms of Γ0(N) ⊂ Sp(2,ℤ) has a very simple unified structure for N = 1, 2, 3, 4, taking Neben-type case (the case with character) for N = 3 and 4. All are generated by 5 generators, and all the fifth generators are obtained by using the other four by means of differential operators, and it is also obtained as Borcherds products. As an appendix, examples of Euler factors of L-functions of Siegel modular forms of Sp(2,ℤ) of odd weight are given.
AMSC: Primary 11F46, Secondary 11F60, Secondary 11F50
References
- A. N. Andrianov, Euler products corresponding to Siegel modular forms of genus 2, Russian Math. Surv. 29(3) (1974) 45–116; Uuspekhi Math. Nauk 29(3) (1974) 43–110 . Google Scholar
-
A. N. Andrianov , Quadratic Forms and Hecke Operators ( Springer-Verlag , Berlin, Heidelberg, New York, London, Paris, Tokyo , 1987 ) . Crossref, Google Scholar - Invent. Math. 120(1), 161 (1995). Crossref, ISI, Google Scholar
-
J. Bruinier , Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors ,Lecture Notes in Mathematics 1780 ( Springer-Verlag , Berlin , 2002 ) . Crossref, Google Scholar - J. Number Theory 68(2), 160 (1998). Crossref, ISI, Google Scholar
-
M. Eichler and D. Zagier , The Theory of Jacobi Forms ( Birkhäuser, Boston, Basel, Stittgart , 1985 ) . Crossref, Google Scholar - Int. J. Math. 9(4), 443 (1998). Link, ISI, Google Scholar
- Manuskripte Forschergruppe Arith. Mannheim–Heidelberg 6, 1 (2002). Google Scholar
- Int. J. Math. 9(2), 201 (1998). Link, ISI, Google Scholar
- J. Math. Soc. Japan 56(2), 375 (2004). Crossref, ISI, Google Scholar
- S. Hayashida and T. Ibukiyama, Siegel modular forms of half integral weights and a lifting conjecture, preprint . Google Scholar
- Int. J. Math. 2(1), 17 (1991). Link, Google Scholar
- Commun. Math. Univ. St. Pauli 48, 103 (1999). ISI, Google Scholar
- T. Ibukiyama, Vector valued Siegel modular forms of symmetric tensor representation of degree two, preprint (2000) . Google Scholar
- T. Ibukiyama, Vector valued Siegel modular forms of Sym(4) and Sym(6), preprint (2001) . Google Scholar
- T. Ibukiyama, Vector valued Siegel modular forms of half integral weight, preprint (2004) . Google Scholar
- J. Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962) 175–200; II, ibid. 86 (1964) 392–412 . Google Scholar
- Amer. J. Math. 89, 817 (1967). Crossref, ISI, Google Scholar
