Theta series for the Leech lattice

Question

The Leech lattice, found by John Leech in 1965, is a fascinating combinatorial object and gives the best possible lattice sphere packing in $\mathbb{R}^{24}$. This result was proved by Cohn and Kumar in a paper published in the Annals in 2009. In that paper, they conjectured

"We conjecture that this method can be used to recover each of the coefficients of the Leech lattice's theta series..."

The theta series for a lattice $\Lambda$ is defined by

$$\displaystyle \Theta_\Lambda(z) = \sum_{x \in \Lambda} q^{x \cdot x},$$

where $q = e^{2 \pi i z}$ and $z$ is in the upper half-plane. The Theta series for the Leech lattice $\Lambda_{24}$ is known to be

$$\displaystyle \Theta_{\Lambda_{24}}(z) = \sum_{x \in \Lambda} \frac{65520}{691} (\sigma_{11}(m) - \tau(m)) q^{2m},$$

where $\sigma_{11}(n) = \sum_{d | n} d^{11}$ and $\tau(n)$ is the Ramanujan tau function.

I am wondering what is the status of this conjecture, and whether someone has confirmed it either way?

Thanks for any insights.

Answer 1

That sentence in the paper is actually a bit vague, in that it doesn't explain exactly what "this method" consists of. There are several more or less plausible interpretations, depending on how closely one wants to imitate what's done in the paper. As far as I know, all of them are open above one dimension.

The least demanding interpretation is as follows. Suppose we have found an auxiliary function that solves the $24$-dimensional sphere packing problem exactly using the LP bounds, and that furthermore pins down the vector lengths of the optimal lattice and its dual to be of the form $\sqrt{2k}$ for $k>1$. Then if we want the theta series coefficients, we can forget about inequalities and just work with equalities. We need functions that zero out all but a few terms in Poisson summation. For example, we could ask for radial functions $g_i \colon \mathbb{R}^{24} \to \mathbb{R}$ such that $g$ vanishes at radius $\sqrt{2k}$ for $k>i$ but not $k=i$, while $\widehat{g}$ vanishes at radius $\sqrt{2k}$ for $k \ge i$. Then Poisson summation using $g_2$ would suffice to determine the number of minimal vectors in terms of $g_2(0)$ and $\widehat{g_2}(0)$, using $g_3$ would give the number of vectors of norm $6$ in terms of $g_3(0)$, $\widehat{g_3}(0)$, $g_3(2)$, and $\widehat{g_3}(2)$ (once we know the number of minimal vectors), etc. Such functions almost certainly exist, but I don't know how to write them down offhand.

More demanding interpretations impose inequalities, as in the example from the paper, so that they can be applied even if you don't know the possible vector lengths, but rather just some information about them. For example, in the paper we knew that the nonzero vector lengths were roughly $2$ or at least roughly $\sqrt{6}$. This is quite a bit more subtle, but not so useful: aside from the case in the paper, I don't know of any plausible scenario in which you would have this kind of knowledge but wouldn't know even more. (It doesn't make sense to apply these methods iteratively, since if done properly they should immediately give you enough information to pass to the less demanding case listed above on the next iteration.) But you can still write down natural generalizations of what's done in the paper, which would let you compute each theta series coefficient once you know all the previous coefficients. At least the first few seem to work numerically, but I don't know how to analyze them.