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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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##### References:
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