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The Eichler Commutation Relation for theta series with spherical harmonics

Lynne H. Walling

Acta Arithmetica (1993)

  • Volume: 63, Issue: 3, page 233-254
  • ISSN: 0065-1036

Abstract

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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.

How to cite

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Lynne H. Walling. "The Eichler Commutation Relation for theta series with spherical harmonics." Acta Arithmetica 63.3 (1993): 233-254. <http://eudml.org/doc/206519>.

@article{LynneH1993,
abstract = { 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. },
author = {Lynne H. Walling},
journal = {Acta Arithmetica},
keywords = {Eichler commutation relation; integral weight; theta series with spherical harmonics; Hilbert modular cusp forms; half-integral weight; Hecke operators; Hilbert modular forms},
language = {eng},
number = {3},
pages = {233-254},
title = {The Eichler Commutation Relation for theta series with spherical harmonics},
url = {http://eudml.org/doc/206519},
volume = {63},
year = {1993},
}

TY - JOUR
AU - Lynne H. Walling
TI - The Eichler Commutation Relation for theta series with spherical harmonics
JO - Acta Arithmetica
PY - 1993
VL - 63
IS - 3
SP - 233
EP - 254
AB - 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.
LA - eng
KW - Eichler commutation relation; integral weight; theta series with spherical harmonics; Hilbert modular cusp forms; half-integral weight; Hecke operators; Hilbert modular forms
UR - http://eudml.org/doc/206519
ER -

References

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  1. [1] A. N. Andrianov, Quadratic Forms and Hecke Operators, Springer, New York 1987. Zbl0613.10023
  2. [2] M. Eichler, The Basis Problem for Modular Forms and the Traces of the Hecke Operators, Lecture Notes in Math. 320, Springer, 1973. 
  3. [3] M. Eichler, On theta functions of real algebraic number fields, Acta Arith. 33 (1977), 269-292. Zbl0348.10017
  4. [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). Zbl0689.10034
  5. [5] O. T. O'Meara, Introduction to Quadratic Forms, Springer, New York 1973. 
  6. [6] T. R. Shemanske and L. H. Walling, Twists of Hilbert modular forms, Trans. Amer. Math. Soc., to appear. Zbl0785.11029
  7. [7] L. H. Walling, Hecke operators on theta series attached to lattices of arbitrary rank, Acta Arith. 54 (1990), 213-240. Zbl0644.10023
  8. [8] L. H. Walling, On lifting Hecke eigenforms, Trans. Amer. Math. Soc. 328 (1991), 881-896. Zbl0749.11028
  9. [9] L. H. Walling, Hecke eigenforms and representation numbers of quadratic forms, Pacific J. Math. 151 (1991), 179-200. Zbl0749.11027
  10. [10] L. H. Walling, Hecke eigenforms and representation numbers of arbitrary rank lattices, Pacific J. Math., 156 (1992), 371-394. Zbl0738.11037
  11. [11] L. H. Walling, An arithmetic version of Siegel's representation formula, to appear. 
  12. [12] L. H. Walling, A remark on differences of theta series, J. Number Theory, to appear. Zbl0810.11026

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