Worked solution: Viazovska's modular form solution to sphere packing in dimensions 8 and 24

For the Leech lattice $\Lambda_{24}$, the minimal vector length is $2$, so the required Cohn–Elkies radius is $r=2$. Following exactly the template of Step 5, Cohn, Kumar, Miller, Radchenko, and Viazovska (2016) build a $+1$-eigenfunction from a weight $-8$, depth $2$ weakly holomorphic quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and a $-1$-eigenfunction from a weight $-10$ weakly holomorphic modular form for the congruence subgroup $\Gamma(2)$, engineered so that both pieces have double zeros at every vector length $\sqrt{2n}$ of $\Lambda_{24}$ for $n \ge 2$. The same Poisson-summation argument as in Step 3, now for $\Lambda_{24}$, shows the resulting combination $f=\varphi^{+}+\varphi^{-}$ achieves equality in the Cohn–Elkies bound, so $\Delta_{24} = \pi^{12}/12! \approx 0.0019296$ and, by the same uniqueness argument as Step 6, the Leech lattice is the unique periodic packing of that density in $\mathbb{R}^{24}$.

The paper was posted on arXiv only one week after Viazovska's single-author $d=8$ paper and was likewise published in the Annals of Mathematics in 2017; the authors also confirmed several special-value conjectures about the auxiliary function that Cohn and Miller had proposed in advance, evidence that the modular-forms mechanism, not coincidence, is what makes both dimensions work.