The theta functions of sublattices of the Leech lattice. (English) Zbl 0579.10010

Kondo, Takeshi; Tasaka, Takashi

Nagoya Math. J. 101, 151-179 (1986).

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)

Keywords:

Leech lattice; Mathieu group; Dedekind eta-function; moonshine of Fischer-Griess’ monster; identities between Jacobi theta functions

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