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Atkin-Lehner eigenforms and strongly modular lattices. (English) Zbl 0898.11014
There are two arithmetical objects associated with any triple \((k,\ell,\chi)\) consisting of an even positive integer \(k\), a squarefree positive integer \(\ell\), and a character \(\chi\) of the group of Atkin-Lehner involutions on \(\Gamma_0(\ell)\) which maps the Fricke involution to \((-1)^{k/2}\). Namely, there is a space of common Atkin-Lehner eigenforms of weight \(k\), and there is a genus of positive definite lattices in dimension \(2k\). In general, Siegel’s weighted mean of the theta series from the latter genus lies in the former space. For an individual lattice, however, the same holds under the condition of strong modularity introduced in this paper. Many interesting lattices known in higher-dimensional Euclidean space are strongly modular, and their theta series are explained by the theory of modular forms.

11F27 Theta series; Weil representation; theta correspondences
11E12 Quadratic forms over global rings and fields
11F20 Dedekind eta function, Dedekind sums
11H55 Quadratic forms (reduction theory, extreme forms, etc.)