Hecke eigenforms and representation numbers of arbitrary rank lattices

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Introduction. Given a totally positive quadratic form Q over a totally real number field K, one can obtain a Hilbert modular form by restricting Q to a lattice L and forming the theta series attached to L; the Fourier coefficients of the theta series are the representation numbers of Q on L. The space of Hilbert modular forms generated by all theta series attached to lattices of the same weight, level and character is invariant under a subalgebra of the Hecke algebra, hence one can (in theory) diagonalize this space of modular forms with respect to an appropriate Hecke subalgebra and infer relations on the representation numbers of the lattices. In a previous paper the author found such relations by constructing eigenforms from theta series attached to lattices of even rank which are “nice” at dyadic primes; the purpose of this paper is to extend the previous results to all lattices. We begin by proving a lemma (Lemma 1.1) which allows us to remove the restriction regarding dyadic primes. Then using our previous work we find that associated to any even rank lattice L is a family of lattices famL which is partitioned into nuclear families (which are genera when the ground field is Q), and the averaged representation numbers of these nuclear families satisfy arithmetic relations (Theorem 1.2). In §2 we define “Fourier coefficients” attached to integral ideals for a half-integral weight Hilbert modular form. Then in analogy to the case K = Q, we describe the effect of the Hecke operators on these Fourier coefficients (Theorem 2.5). In §3 we use theta series attached to odd rank lattices to construct eigenforms for the Hecke operators; the results of §2 then give us arithmetic relations on the representation numbers of the odd rank lattices. When the ground field is Q, we may assume Q(L) ⊆ Z and then these relations may be stated as