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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
Full Text: DOI

Online Encyclopedia of Integer Sequences:

Theta series of 32-dimensional Quebbemann lattice Q_32.
Theta series of 28-dimensional Quebbemann lattice.
Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.
Theta series of hypothetical extremal 3-modular even 36-dimensional lattice with minimal norm 8 and det = 3^18.
Theta series of hypothetical extremal 3-modular even 48-dimensional lattice with minimal norm 10 and det = 3^24.
Theta series of (putative) extremal 2-modular even lattice in dimension 20.
Theta series of (putative) extremal 2-modular even lattice in dimension 24.
Theta series of (putative) extremal 2-modular even lattice in dimension 28.
Theta series of (putative) extremal 2-modular even lattice in dimension 36.
Theta series of (putative) extremal 2-modular even lattice in dimension 40.
Theta series of (putative) extremal 2-modular even lattice in dimension 44.
Theta series of (putative) extremal 2-modular even lattice in dimension 48.
Theta series of (putative) extremal 2-modular even lattice in dimension 52.
Theta series of (putative) extremal 2-modular even lattice in dimension 56.
Theta series of (putative) extremal 2-modular even lattice in dimension 60.
Theta series of (putative) extremal 2-modular even lattice in dimension 64.
Theta series of (putative) extremal 2-modular even lattice in dimension 68.
Theta series of (putative) extremal 2-modular even lattice in dimension 72.
Theta series of (putative) extremal 2-modular even lattice in dimension 76.
Theta series of (putative) extremal 2-modular even lattice in dimension 80.
Theta series of (putative) extremal 2-modular even lattice in dimension 84.
Theta series of (putative) extremal 2-modular even lattice in dimension 88.
Theta series of (putative) extremal 2-modular even lattice in dimension 92.
Theta series of (putative) extremal 2-modular even lattice in dimension 96.
Theta series of extremal 3-modular even lattice in dimension 18.
Theta series of extremal 3-modular even lattice in dimension 20.
Theta series of extremal 3-modular even lattice in dimension 22.
Theta series of extremal 3-modular even lattice in dimension 26.
Theta series of extremal 3-modular even lattice in dimension 28.
Theta series of extremal 3-modular even lattice in dimension 30.
Theta series of extremal 3-modular even lattice in dimension 32.
Theta series of extremal 3-modular even lattice in dimension 34.
Theta series of (putative) extremal 3-modular even lattice in dimension 38.
Theta series of extremal 3-modular even lattice in dimension 40.
Theta series of (putative) extremal 3-modular even lattice in dimension 42.
Theta series of (putative) extremal 3-modular even lattice in dimension 44.
Theta series of (putative) extremal 3-modular even lattice in dimension 46.
Theta series of (putative) extremal 3-modular even lattice in dimension 50.
Theta series of (putative) extremal 3-modular even lattice in dimension 52.
Theta series of (putative) extremal 3-modular even lattice in dimension 54.
Theta series of (putative) extremal 3-modular even lattice in dimension 56.
Theta series of (putative) extremal 3-modular even lattice in dimension 58.
Theta series of (putative) extremal 3-modular even lattice in dimension 60.
Theta series of (putative) extremal 3-modular even lattice in dimension 62.
Theta series of (putative) extremal 3-modular even lattice in dimension 64.
Theta series of (putative) extremal 3-modular even lattice in dimension 66.
Theta series of (putative) extremal 3-modular even lattice in dimension 68.
Theta series of (putative) extremal 3-modular even lattice in dimension 70.
Theta series of (putative) extremal 3-modular even lattice in dimension 72.
Theta series of (putative) extremal 3-modular even lattice in dimension 74.
Theta series of (putative) extremal 3-modular even lattice in dimension 76.
Theta series of (putative) extremal 3-modular even lattice in dimension 78.
Theta series of (putative) extremal 3-modular even lattice in dimension 80.
Theta series of (putative) extremal 3-modular even lattice in dimension 82.
Theta series of (putative) extremal 3-modular even lattice in dimension 84.
Theta series of (putative) extremal 3-modular even lattice in dimension 86.
Theta series of (putative) extremal 3-modular even lattice in dimension 88.
Theta series of (putative) extremal 3-modular even lattice in dimension 90.
Theta series of (putative) extremal 3-modular even lattice in dimension 92.
Theta series of (putative) extremal 3-modular even lattice in dimension 94.
Theta series of (putative) extremal 3-modular even lattice in dimension 96.