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The number $65520$ arises in two very different scenarios:

  1. It occurs in the formula for the theta series of the Leech lattice:

$$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{65520}{691}\left(\sigma_{11}(m) - \tau(m) \right) q^m $$

  1. The order of the image of the J-homomorphism in the stable homotopy group $\pi_{23}^S$ is also $65520$.

A more concrete version of the first scenario is to note that $65520$ is exactly the greatest common divisor of the coefficients of the non-constant terms in the theta series:

$$ \gcd(196560,16773120,398034000,4629381120, 34417656000, \dots) = 65520 $$

Equivalently, it's the greatest common divisor, over all origin-centred spherical shells in the Leech lattice, of the number of lattice vectors in that shell. For brevity, we'll write this as $\gcd(\Lambda_{24}) = 65520$.

We can try this with various other lattices to see whether they match the order of the image of the J-homomorphism in the stable homotopy group $\pi_{n-1}^S$, and in many cases they do:

  • $\gcd(D_4) = 24 = |J(\pi_3)|$
  • $\gcd(E_8) = 240 = |J(\pi_7)|$
  • $\gcd(K_{12}) = 252$ but $|J(\pi_{11})| = 504$ (mismatch!)
  • $\gcd(BW_{16}) = 480 = |J(\pi_{15})|$
  • $\gcd(\Lambda_{24}) = 65520 = |J(\pi_{23})|$
  • $\gcd(P_{48p}) = 131040 = |J(\pi_{47})|$
  • $\gcd(\Lambda_{72}) = 138181680 = |J(\pi_{71})|$

Note that the 12-dimensional Coxeter-Todd lattice is a counterexample to this pattern; its $\gcd$ is a factor of 2 smaller than the order of the image of the J-homomorphism.

The last three of these are 'extremal Type II unimodular lattices', which can only occur in dimensions $n$ which are divisible by 24. Note that the theta series of an extremal unimodular lattice of dimension $n$ is independent of the lattice; for example, there are a few non-isomorphic extremal lattices in dimension 48, and they all necessarily have the same theta series. This suggests the following well-defined conjecture:

Conjecture: The $\gcd$ of the coefficients of the non-constant terms in the theta series for an extremal unimodular lattice of dimension $24k$ is precisely the order of the image of the J-homomorphism in the stable homotopy group $\pi_{24k-1}^S$.

Can this conjecture be proved? More generally, why should the theta series of highly symmetrical lattices in dimension $n$ be connected to the order of the image of the J-homomorphism in the stable homotopy group $\pi_{n-1}^S$?

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    $\begingroup$ The order of im J in $\pi_{24k-1}^S$ is the denominator of the Bernoulli number $B_{12k}/24k$, and $B_n/n$ shows up as the only term in Eisenstein series that isn't a sums-of-divisors function. You can see from your formula how the theta-series decomposes into coefficients of the Eisenstein $E_{12}$ and the cusp form $\Delta$. The relation between the theta-series coefficients of a lattice and im J just comes about because of how the theta-series decomposes in terms of modular forms, and in particular what multiple of the Eisenstein series in the relevant weight appears in the decomposition. $\endgroup$
    – user164898
    Commented Aug 8, 2022 at 15:42
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    $\begingroup$ A direct connection between $\pi_*^S$ and the theta-series of lattices would be terrific and fascinating, by the way! But it seems much easier to believe that any connection between im J and theta-series of lattices simply occurs due to properties of Bernoulli numbers. $\endgroup$
    – user164898
    Commented Aug 8, 2022 at 15:51
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    $\begingroup$ I think I can prove that $|J(\pi_{24k})|$ divides the $\operatorname{gcd}$. The theta series can be decomposed as the sum of an Eisenstein series $E_{12k}=1 + \frac{24k}{B_{12k}} \sum_{n>0}a_nq^n$, $a_n \in \mathbb{Z}$, and the cusp forms $A_1 \Delta \Pi_1, A_2 \Delta^2 \Pi_2, \ldots, A_k \Delta^k$, where $A_i \in \mathbb{Q}$ and $\Pi_i=1+\ldots$ is some linear combination of products of Eisenstein series of appropriate weights. It's easy to see that the cusp forms $\Delta^i \Pi_i$ can be simultaneously chosen such that all coefficients are integers and the first nonzero coefficient is $1$. $\endgroup$
    – pregunton
    Commented Aug 13, 2022 at 10:58
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    $\begingroup$ E.g. if $k=2$, the sum of cusp forms is $A_1\Delta (a E_4^3+b E_6^2) + A_2 \Delta^2$ for $a+b=1$, which can be re-expressed as $A_1 \Delta E_4^3 + A_2' \Delta^2)$ where $A_2' = A_2-1728b$. But since the theta series corresponds to an extremal lattice, one needs to succesively kill the $a_i q^i$ terms for $i=1, \ldots, k$. So an easy induction shows that $A_i = \text{integer} \times \frac{24k}{B_{12k}}$, which implies that all remaining nonconstant coefficients of the theta series are necessarily multiples of the numerator of $\frac{24k}{B_{12k}}$, that is, multiples of $|J(\pi_{24k})|$. $\endgroup$
    – pregunton
    Commented Aug 13, 2022 at 11:00

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