Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Computations of Siegel modular forms of genus two


Author: Nils-Peter Skoruppa
Journal: Math. Comp. 58 (1992), 381-398
MSC: Primary 11F46; Secondary 11F60, 11F66
DOI: https://doi.org/10.1090/S0025-5718-1992-1106982-0
MathSciNet review: 1106982
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We explain the basic notions and theorems for doing computations in the theory of Siegel modular forms of degree two, on the full modular group and of even weight. This synopsis concludes with a handy and computationally realistic algorithm for tabulating the Fourier coefficients of such forms and the Euler factors of their Spinor zeta functions. In the second part of this paper we present and discuss some of the results of actual computations which we performed following this algorithm. We point out two (theoretically) striking phenomena that are implied by the results of these computations.


References [Enhancements On Off] (What's this?)

  • [1] A. N. Andrianov, Quadratic forms and Hecke operators, Springer, Berlin, 1987. MR 884891 (88g:11028)
  • [2] -, Euler products corresponding to Siegel modular forms of genus 2, Russian Math. Surveys 29 (1974), no. 3, 45-116.
  • [3] C. Batut, D. Bernardi, H. Cohen, and M. Olivier, PARI-GP, Université Bordeaux 1, Bordeaux, 1989.
  • [4] H. Cohen, N.-P. Skoruppa, and D. Zagier, Tables of modular forms, Springer-Verlag, 1992 (to appear). MR 2432111
  • [5] M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, 1985. MR 781735 (86j:11043)
  • [6] E. Freitag, Siegelsche Modulfunktionen, Springer, Berlin, 1983. MR 871067 (88b:11027)
  • [7] J.-I. Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175-200. MR 0141643 (25:5040)
  • [8] H. Klingen, Zum Darstellungssatz für Siegelsche Modulformen, Math. Z. 102 (1967), 20-43; Berichtigung, Math. Z. 105 (1967), 399-400. MR 0219473 (36:2555)
  • [9] W. Kohnen and N.-P. Skoruppa, A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math. 95 (1989), 541-558. MR 979364 (90b:11050)
  • [10] N. Kurokawa, Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Invent. Math. 49 (1978), 149-165. MR 511188 (80b:10040)
  • [11] H. Maaß, Über eine Spezialschar von Modulformen zweiten Grades. I, II, III, Invent. Math. 52 (1979), 95-104; 53 (1979), 249-253, 255-265. MR 532746 (80f:10031)
  • [12] T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256 (1981), 323-340. MR 626953 (83c:10034)
  • [13] N.-P. Skoruppa, Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts, Bonner Math. Schriften, no. 159, Univ. Bonn, Bonn, 1985. MR 806354 (87a:11045)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11F46, 11F60, 11F66

Retrieve articles in all journals with MSC: 11F46, 11F60, 11F66


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1992-1106982-0
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society