Hecke eigenforms and representation numbers of arbitrary rank lattices
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 \(L\). In a previous paper the author found relations on these representation numbers when the rank of the lattice \(L\) is even and \(L\) is ``nice'' at dyadic primes. In this paper we extend the previous results to all lattices \(L\). To do this, we define ``Fourier coefficients'' attached to integral ideals for half-integral weight Hilbert modular forms, and we describe the effect of the Hecke operators on these Fourier coefficients. We then use theta series to construct an eigenform for the Hecke operators. When the ground field is \(\mathbb{Q}\), we may assume \(Q(L)\subseteq\mathbb{Z}\) and then these relations may be stated as \[ \begin{gathered} {\mathfrak r}(\hbox{gen }L,2p^ 2a)=\\(1-p^{(m-3)/2}\chi_ L(p)(-1\mid p)^{(m-1)/2}(2a\mid p)+p^{m-2}){\mathfrak r}(\hbox{gen }L,2a)-p^{m-2}{\mathfrak r}(\hbox{gen }L,2a/p^ 2)\end{gathered} \] where \({\mathfrak r}(\hbox{gen }L,2a)\) is the average number of times the lattices in the genus of \(L\) represent \(2a\), \(m\) is the rank of \(L\), \(p\) is a prime not dividing the level of \(L\), and \(\chi_ L\) is the character attached to \(L\).
half-integral weight Hilbert modular forms
scientific article; zbMATH DE number 4216100