Eichler commutation relations for theta series of nebentypus
type: Item
instance of: scholarly article
title: Eichler commutation relations for theta series of nebentypus
zbMATH Open document ID: 0627.10018
DOI: 10.1007/BF01450086
author: Paul Ponomarev
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164394
review text: From the work of Eichler we have the fundamental principle that the effect of a Hecke operator \(T(m)\) acting on a suitable set of elliptic modular theta series can be represented by an ``Anzahlmatrix'' provided \(m\) is the norm of a similitude of the underlying quadratic space. If \(m\) is not the norm of a similitude no such ``Eichler commutation relations'' are known. We consider the case of theta series of weight two, prime level \(p\equiv 1\pmod 4\) and quadratic character \((\cdot/p)\). These come from positive definite integral quaternary quadratic forms of discriminant \(p\) and their adjoints of discriminant \(p^3\). In this situation we have an Eichler commutation relation for \(T(m)\) as long as \(m\) is a norm from \(K=\mathbb{Q}(\sqrt{p})\). The Anzahlmatrices can be expressed in terms of Brandt matrices associated to the definite quaternion algebra of discriminant 1 over \(K\). In addition, we have the standard action of \(W=\left[ \begin{matrix} 0 & -1\\ p & 0\end{matrix} \right]\) corresponding to the theta inversion formula. We consider the question of whether the resulting ``Brandt matrix representation'' can be extended to the algebra generated by \(W\) and all Hecke operators. Theorem 1 states that a necessary condition for such an extension to exist is that \(\pm \sqrt{p}\) not be an eigenvalue of \(T(p)\). Theorem 2 gives a partial converse to Theorem 1: The representation is extendable if \(\pm 2\sqrt{p}\) is not an eigenvalue of \(B(\mathfrak p)\), the Brandt matrix associated to the ramified prime \(\mathfrak p\) of \(K\). We note that the condition in Theorem 1 is also Waldspurger's criterion for a positive solution to the basis problem.
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 4019154
zbMATH Keywords: Anzahlmatrix
zbMATH Keywords: theta series of weight two
zbMATH Keywords: integral quaternary quadratic forms
zbMATH Keywords: Eichler commutation relation
zbMATH Keywords: quaternion algebra
zbMATH Keywords: Brandt matrix representation
zbMATH Keywords: Hecke operator
zbMATH Keywords: Waldspurger's criterion
zbMATH Keywords: basis problem
MaRDI profile type: Publication
cites work: Q5816776
cites work: Q4196623
cites work: Q5605180
cites work: Class numbers of quadratic forms over real quadratic fields
cites work: On the coincidence of two Dirichlet series associated to cusp forms of Hecke's ''Neben''-type and Hilbert modular forms over a real quadratic field
cites work: Ternary quadratic forms and Shimura’s correspondence
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description: scientific article; zbMATH DE number 4019154