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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

##### MSC:
 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)
##### Keywords:
quadratic forms; Leech lattice; Jacobi theta functions