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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.

##### MSC:
 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
##### Keywords:
modular lattices; modular group; Fricke group; automorphic form
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