Separators of Siegel modular forms of degree two

Heim, Bernhard

doi:10.1090/S0002-9939-08-09597-X

Separators of Siegel modular forms of degree two
HTML articles powered by AMS MathViewer

by Bernhard Heim PDF

Proc. Amer. Math. Soc. 136 (2008), 4167-4173

Abstract:

We prove that cuspidal Siegel modular forms of degree two and weight $2k$ are uniquely determined by their Fourier coefficients on small subsets of matrices of content one. This generalizes results of Breulmann, Kohnen, Katsurada, Scharlau and Walling. We give applications to the space of Saito-Kurokawa lifts.

References

A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
Stefan Breulmann and Winfried Kohnen, Twisted Maaß-Koecher series and spinor zeta functions, Nagoya Math. J. 155 (1999), 153–160. MR 1711371, DOI 10.1017/S0027763000007029
William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. MR 337789, DOI 10.1007/BF01428197
B. Heim: On the Spezialschar of Maass. arXiv: 0801.1804v1 [math.NT].
H. Katsurada, On the coincidence of Hecke-eigenforms, Abh. Math. Sem. Univ. Hamburg 70 (2000), 77–83. MR 1809535, DOI 10.1007/BF02940903
Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo $l$, Ann. of Math. (2) 147 (1998), no. 2, 453–470. MR 1626761, DOI 10.2307/121015
Cris Poor and David S. Yuen, Linear dependence among Siegel modular forms, Math. Ann. 318 (2000), no. 2, 205–234. MR 1795560, DOI 10.1007/s002080000083
C. S. Rajan, Refinement of strong multiplicity one for automorphic representations of $\textrm {GL}(n)$, Proc. Amer. Math. Soc. 128 (2000), no. 3, 691–700. MR 1707005, DOI 10.1090/S0002-9939-99-05616-6
C. S. Rajan, On strong multiplicity one for $l$-adic representations, Internat. Math. Res. Notices 3 (1998), 161–172. MR 1606395, DOI 10.1155/S1073792898000142
Richard Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. II, Invent. Math. 116 (1994), no. 1-3, 619–643. MR 1253207, DOI 10.1007/BF01231575
Rudolf Scharlau and Lynne H. Walling, A weak multiplicity-one theorem for Siegel modular forms, Pacific J. Math. 211 (2003), no. 2, 369–374. MR 2015741, DOI 10.2140/pjm.2003.211.369
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
D. Zagier, Sur la conjecture de Saito-Kurokawa (d’après H. Maass), Seminar on Number Theory, Paris 1979–80, Progr. Math., vol. 12, Birkhäuser, Boston, Mass., 1981, pp. 371–394 (French). MR 633910