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The Eichler Commutation Relation for theta series with spherical harmonics

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It is well known that classical theta series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of theta series attached to lattices in a fixed genus can yield both cusp forms and Eisenstein series whose weight is one-half the rank of the quadratic form. In contrast, generalized theta series - those augmented with a spherical harmonic polynomial - will always yield cusp forms whose weight is increased by the degree of the spherical harmonic. A recent demonstration of the far-reaching importance of generalized theta series is Hijikata, Pizer and Shemanske's solution to Eichler's Basis Problem [4] (cf. [2]) in which character twists of such theta series are used to provide a basis for the space of newforms.
In this paper we consider theta series with spherical harmonics over a totally real number field. We show that such theta series are Hilbert modular cusp forms whose weight is integral or half-integral, depending on the rank of the associated lattice. We explicitly describe the action of the Hecke operators on these theta series in terms of other theta series, yielding a generalization of the well-known Eichler Commutation Relation. Finally, we use these theta series to construct Hilbert modular forms which are invariant under a subalgebra of the Hecke algebra. We are able to show that if the quadratic form has rank m and the spherical harmonic has degree l, then the theta series attached to the genus of a lattice is identically zero whenever l is small relative to m; in particular, the associated collection of theta series are linearly dependent.
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  • Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, U.S.A.
  • [1] A. N. Andrianov, Quadratic Forms and Hecke Operators, Springer, New York 1987.
  • [2] M. Eichler, The Basis Problem for Modular Forms and the Traces of the Hecke Operators, Lecture Notes in Math. 320, Springer, 1973.
  • [3] M. Eichler, On theta functions of real algebraic number fields, Acta Arith. 33 (1977), 269-292.
  • [4] H. Hijikata, A. K. Pizer and T. R. Shemanske, The basis problem for modular forms on Γ₀(N), Mem. Amer. Math. Soc. 418 (1989).
  • [5] O. T. O'Meara, Introduction to Quadratic Forms, Springer, New York 1973.
  • [6] T. R. Shemanske and L. H. Walling, Twists of Hilbert modular forms, Trans. Amer. Math. Soc., to appear.
  • [7] L. H. Walling, Hecke operators on theta series attached to lattices of arbitrary rank, Acta Arith. 54 (1990), 213-240.
  • [8] L. H. Walling, On lifting Hecke eigenforms, Trans. Amer. Math. Soc. 328 (1991), 881-896.
  • [9] L. H. Walling, Hecke eigenforms and representation numbers of quadratic forms, Pacific J. Math. 151 (1991), 179-200.
  • [10] L. H. Walling, Hecke eigenforms and representation numbers of arbitrary rank lattices, Pacific J. Math., 156 (1992), 371-394.
  • [11] L. H. Walling, An arithmetic version of Siegel's representation formula, to appear.
  • [12] L. H. Walling, A remark on differences of theta series, J. Number Theory, to appear.
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