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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    $\begingroup$ Why is $Z_{RR}(L)$ weight $r$ rather than $r/2$ as one would usually have for the theta function of a rank $r$ lattice? $\endgroup$ – Jeff Harvey Mar 21 '18 at 23:38
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    $\begingroup$ Also, I think a modular form for the full modular group is usually described as having level 1, not 0; and the exponent in the product formula for $\Delta$ is 24, not 12. $\endgroup$ – Noam D. Elkies Mar 22 '18 at 0:32
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    $\begingroup$ I can show that for a "3-framed" lattice your function $Z_{RR}$ vanishes unless $r = 12k$ in which case $Z_{RR}$ is a multiple of $\Delta^{k/2}$ (not $\Delta^k$). For $r=24$, it's the difference between the theta functions of two Niemeier lattices, and thus a multiple of $24\Delta$ (this works for any $L$, regardless of 3-framing). For $r=36$ and beyond I see no reason for the coefficients to be a multiple of anything bigger than the $2$ forced by central symmetry of $L_\pm$. $\endgroup$ – Noam D. Elkies Mar 22 '18 at 3:26
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    $\begingroup$ I trust @NoamD.Elkies more than I trust myself: he confirms the integrality properties asserted in my question (except for $\Delta \leadsto \Delta^{1/2}$). I haven't yet read math.berkeley.edu/~reb/papers/on2/on2.pdf, but arxiv.org/pdf/math/0212397.pdf quotes from it that for $L$ of rank divisible by $24$, $\Theta_L \in 24 \Delta^{r/24} \mathbb Z + c_4\mathbb Z[c_4,\Delta]$. That might complete the proof. $\endgroup$ – Theo Johnson-Freyd Mar 22 '18 at 3:35
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    $\begingroup$ @NoamD.Elkies Hrm. There is a chance that the congruence holds more generally. The 24m+12 case isn't that far from the 24m case. Assuming I understand things correctly, the nontrivial fact about ternary codes is that a certain "index", given by a signed count of code words of maximal Hamming weight, is always divisible by 24. I tried to write it up in this question. $\endgroup$ – Theo Johnson-Freyd Mar 23 '18 at 2:00

The discussion in the comments establishes the conjecture when $r$ is divisible by $24$. When $r$ is merely divisible by $12$, the comments establish that $\Delta^{-r/24} Z_{RR}$ is divisible by $12$. (The latter case follows from the former because $Z_{RR}$ is easily seen to be multiplicative.)

Specifically, when $r$ is divisible by $24$, $Z_{RR}$ is a difference of $\Theta$ series of two even unimodular lattices of rank $r$. Theorem 12.1 of https://math.berkeley.edu/~reb/papers/on2/on2.pdf establishes that every $\Theta$ series of an even unimodular lattice is a polynomial of $\Delta$ and $c_4$, and the coefficient of $\Delta^{r/24}$ is divisible by $24$.

This answer is community wiki in case someone wants to expand it.


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