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?



Niemeier lattices and theta functions
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.


