For the purposes of this question, a rank-$r$ (integral) *lattice* is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It is *even* if $\ell^2 = \langle \ell,\ell\rangle \in 2\mathbb Z$ for all $\ell \in L$ and *odd* if there is some $\ell$ such that $\ell^2$ is odd. Define $L^* = \{ v\in \mathbb R^n \text{ s.t. } \langle v,\ell \rangle \in \mathbb Z \, \forall \ell \in L\}$. $L$ is *unimodular* if $L = L^*$.

Let $L$ be an odd unimodular lattice, and define $L_{ev} = \{\ell \in L \text{ s.t. } \ell^2 \in 2\mathbb Z\}$. Then $L_{ev}$ has index $2$ in $L$, and so index $4$ in $L_{ev}^*$. The cosets of $L_{ev}$ in $L_{ev}^*$ are $L_{ev}$, $L_{odd} := L \smallsetminus L_{ev}$, and two more that I will arbitrarily name $L_+$ and $L_-$. Vectors in $L_\pm$ might have fractional length. (The structure of the group $L_{ev}^*/L_{ev}$ and the lengths of vectors in $L_\pm$ depend on the rank $r$ mod $8$.)

Given a discrete subset $S \subset \mathbb R^r$, the *Theta series* of $S$ is the $q$-series
$ \Theta_S(q) = \sum_{s\in S} q^{s^2/2}. $
Suppose that $L$ is odd and unimodular. I find myself interested in the following function:
$$ Z_{RR}(L) := \Theta_{L_+}(q) - \Theta_{L_-}(q)$$
The letters "$RR$" stand for "Ramond-Ramond", because this this shows up as an RR-sector partition function of a spin conformal field theory built from $L$ (times a power of $\eta$). I'm pretty sure that $Z_{RR}(L)$ is automatically a (level-1) modular form of weight $r$. [Edit: $r/2$.]

Let $\sqrt3\mathbb Z \subset \mathbb R$ denote the (non-unimodular) odd lattice of vectors $\ell$ such that $\ell^2 \in 3\mathbb Z$.
A unimodular lattice $L$ of rank $r$ is *3-framed* if it contains a copy of $(\sqrt3\mathbb Z)^r$. 3-framed lattices are in natural bijection with self-dual ternary codes; Harada and Munemasa used this bijection, together with Borcherds' classification of unimodular lattices of rank $\leq 25$, to classify self-dual ternary codes of rank $24$.

Recall the *discriminant* $\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$. It is a modular form of weight $12$. I'm pretty sure that

**Proposition:** If $L$ is 3-framed and unimodular, then $Z_{RR}(L) \in \mathbb Z[\Delta]$. [Edit: $\mathbb Z[\Delta^{1/2}]$.]

I.e. I am claiming that $Z_{RR}(L)$ vanishes unless $r = 12k$, in which case $\Delta^{-k}Z_{RR}(L) \in \mathbb Z$.

By simply checking against Borcherds' classification, I know that the following conjecture is true in ranks $r = 12$ and $24$:

**Conjecture:** Suppose $L$ is a 3-framed unimodular lattice of rank $r = 12k$. Then the integer $\Delta^{-k}Z_{RR}(L)$ is divisible by $24$. [Edit: $\Delta^{-k/2}$.]

Is the conjecture true in general? Is it known?

[Edit: And please continue to point out errors in the comments — I'm sure there are more.]

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