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I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice gave you this theta function in the first place?

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up vote 18 down vote accepted

The theta series for a Niemeier lattice determines the lattice in most cases, but there are five ambiguous pairs.

The theta series of an even unimodular lattice must be a polynomial in the theta series of $E_8$ and $\Lambda_{24}$ (this is a modular forms calculation). Thus, for Niemeier lattices, it must be a linear combination of those for $E_8^3$ and $\Lambda_{24}$. The constant term must be $1$, so there is one remaining degree of freedom. This means the theta series for a Niemeier lattice is determined by how many roots (i.e., vectors of norm $2$) it has. There are five pairs of Niemeier lattices with the same number of roots, but no triples.

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