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.
@article{bwmeta1.element.bwnjournal-article-aav63i3p233bwm,
author = {Lynne H. Walling},
title = {The Eichler Commutation Relation for theta series with spherical harmonics},
journal = {Acta Arithmetica},
volume = {64},
year = {1993},
pages = {233-254},
zbl = {0773.11029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav63i3p233bwm}
}
Lynne H. Walling. The Eichler Commutation Relation for theta series with spherical harmonics. Acta Arithmetica, Tome 64 (1993) pp. 233-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav63i3p233bwm/
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