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The theta functions of sublattices of the Leech lattice. (English) Zbl 0579.10010
Let \(\Lambda\) be the Leech lattice. Define for \(m\in M_{24}\) (Mathieu group) the sublattice \(\Lambda_ m\) of vectors invariant under m, and \(\theta_ m(z)\) the theta function of \(\Lambda_ m\). This theta function is expressed explicitly by the classical Jacobi theta functions \(\vartheta_ i(z)\) \((i=2,3,4)\) and the Dedekind eta-function \(\eta\). The main theorem says that \(\theta_ m(z)/\eta_ m(z)\) \((\eta_ m\) suitably defined via \(\eta)\) are modular functions which appear in a moonshine of Fischer-Griess’ monster. On the way several (new) identities between Jacobi theta functions show up.
Reviewer: M.Peters

MSC:
11F11 Holomorphic modular forms of integral weight
11H06 Lattices and convex bodies (number-theoretic aspects)
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