Hecke eigenforms and representation numbers of arbitrary rank lattices
type: Item
instance of: scholarly article
title: Hecke eigenforms and representation numbers of arbitrary rank lattices
zbMATH Open document ID: 0738.11037
author: Lynne H. Walling
published in: Pacific Journal of Mathematics
publication date: 1992-01-01T00:00:00Z
review text: 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 \(L\).
reviewed by: Lynne H. Walling
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F60
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 4216100
zbMATH Keywords: theta series
zbMATH Keywords: representation numbers
zbMATH Keywords: half-integral weight Hilbert modular forms
zbMATH Keywords: Hecke operators
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: eigenform
MaRDI profile type: Publication
full work available at URL: pjm.1992.156.371
OpenAlex ID: W1973538554
DOI: 10.2140/PJM.1992.156.371
description: scientific article; zbMATH DE number 4216100