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The theta functions of sublattices of the Leech lattice. II. (English) Zbl 0646.10015
Let L be the Leech lattice, let \(2^{12}M_{24}\) be the so-called monomial subgroup of the automorphism group of L (denoted by \(\cdot O)\). Let \(L_{\pi}=\{v\in L|\) \(v\circ \pi =v\}\) for \(\pi\in \cdot O\), \(\theta_{\pi}(z)=\sum_{v\in L_{\pi}}\exp (\pi iz<v,v>)\). An expression for \(\theta_{\pi}\) for \(\pi \in 2^{12}M_{24}\) is given in terms of Jacobi theta functions \(\theta_ i(z)\) (2\(\leq i\leq 4)\) and a question of J. H. Conway and S. P. Norton [Monstrous moonshine, Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)] for functions \(\theta_{\pi}(z)/\eta_{\pi}(z)\) is studied (for \(\pi \in M_{24}\) this was done in the first part of this paper, Nagoya Math. J. 101, 151-179 (1986; Zbl 0579.10010)].
Similar questions and results are contained in a paper of M. Koike [Modular forms and the automorphism group of Leech lattice, Nagoya Math. J. (1988)].
Reviewer: M.Peters

11E16 General binary quadratic forms
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11H06 Lattices and convex bodies (number-theoretic aspects)