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Modular lattices in Euclidean spaces. (English) Zbl 0874.11038
The Fricke group \(\Gamma_x(\ell)\), \(\ell\in\mathbb{N}\), is the extension of the congruence subgroup \(\Gamma_0(\ell)\) of the modular group \(SL(2,\mathbb{Z})\) by the element \(T_\ell=\left(\begin{smallmatrix} 0 & (\sqrt\ell)^{-1}\\ -\sqrt\ell & 0\end{smallmatrix}\right)\). If \(\Lambda\) is an even lattice on an Euclidean \(n\)-space the theta function of \(\Lambda\) is defined as \(\Theta_\Lambda(z)= \sum_{v\in\Lambda}q^{(v\cdot v)/2}\), \(q= e^{2\pi iz}\). If \(n=2k\) and \(\Lambda\) is an even \(\sigma\)-modular lattice for some similarity \(\sigma\) of norm \(\ell\) then \(\Theta_\Lambda(z)\) is an automorphic form of weight \(k\) for \(\Gamma_*(\ell)\). This connection is discussed in detail.

11F06 Structure of modular groups and generalizations; arithmetic groups
11F27 Theta series; Weil representation; theta correspondences
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11F30 Fourier coefficients of automorphic forms
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