Quebbemann, H.-G. Modular lattices in Euclidean spaces. (English) Zbl 0874.11038 J. Number Theory 54, No. 2, 190-202 (1995). 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. Reviewer: G.Rosenberger (Dortmund) Cited in 5 ReviewsCited in 46 Documents 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 × Cite Format Result Cite Review PDF 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.