Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On lifting Hecke eigenforms


Author: Lynne H. Walling
Journal: Trans. Amer. Math. Soc. 328 (1991), 881-896
MSC: Primary 11F41; Secondary 11F27, 11F60
MathSciNet review: 1061779
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A classical Hilbert modular form $ f \in {\mathcal{M}_k}({\Gamma _0}(\mathcal{N},\mathcal{I}),{\chi _\mathcal{N}})$ cannot be an eigenform for the full Hecke algebra. We develop a means of lifting a classical form to a modular form $ F \in { \oplus _\lambda }{\mathcal{M}_k}({\Gamma _0}(\mathcal{N},{\mathcal{I}_\lambda }),{\chi _\mathcal{N}})$ which is an eigenform for the full Hecke algebra. Using this lift, we develop the newform theory for a space of cusp forms $ {\mathcal{S}_k}({\Gamma _0}(\mathcal{N},\mathcal{I}),{\chi _\mathcal{N}})$; we also use theta series to construct eigenforms for the full Hecke algebra.


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

  • [1] Martin Eichler, On theta functions of real algebraic number fields, Acta Arith. 33 (1977), no. 3, 269–292. MR 0563061
  • [2] Erich Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, New York-Berlin, 1981. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. MR 638719
  • [3] Wen Ch’ing Winnie Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315. MR 0369263
  • [4] Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0256993
  • [5] O. T. O'Meara, Introduction to quadratic forms, Springer-Verlag, New York, 1973.
  • [6] T. R. Shemanske and L. H. Walling, Twists of Hilbert modular forms, submitted 1990.
  • [7] Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
  • [8] Goro Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math. (2) 111 (1980), no. 2, 313–375. MR 569074, 10.2307/1971202
  • [9] C. L. Siegel, Über die analytische Theorie der quadratischen Formen, Gesammelte Abhandlungen, Springer-Verlag, New York, 1966, pp. 326-405.
  • [10] -, Über die analytische Theorie der quadratischen Formen III, Gesammelte Abhandlungen, Springer-Verlag, New York, 1966, pp. 469-548.
  • [11] Lynne H. Walling, Hecke operators on theta series attached to lattices of arbitrary rank, Acta Arith. 54 (1990), no. 3, 213–240. MR 1056106

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11F41, 11F27, 11F60

Retrieve articles in all journals with MSC: 11F41, 11F27, 11F60


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1061779-0
Keywords: Hilbert modular forms, newforms, theta series, quadratic forms
Article copyright: © Copyright 1991 American Mathematical Society