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

A lattice is even and unimodular if all vector-length-squared values are even integers and the lattice equals its own dual; such lattices exist only in dimensions divisible by $8$, and in dimensions $8$ and $24$ they are extremal in the technical sense of having the largest possible minimal vector length for their dimension ($E_8$ in $8$, the Leech lattice in $24$). For an even unimodular lattice, the theta series $\Theta_\Lambda(z) = \sum_{v \in \Lambda} q^{|v|^2/2}$ is automatically a modular form of weight $\dim(\Lambda)/2$ for $\mathrm{SL}_2(\mathbb{Z})$; extremality then pins this modular form down to an essentially unique choice built from the ring of modular forms (for $E_8$, $\Theta_{E_8}=E_4$; for the Leech lattice, $\Theta_{\Lambda_{24}} = E_4^3 - 720\,\Delta$), which is precisely the rigidity that lets Viazovska's contour-integral construction hit every required vector length with a double zero.

Cohn and Elkies had conjectured back in 2003 that magic functions should exist for $E_8$ and the Leech lattice specifically, based on how astonishingly tight their numerical linear programming bounds already were; Viazovska's contribution was to find the exact analytic mechanism (modular forms) realising that conjecture. No comparably rigid extremal lattice is known in other dimensions, so whether some other technique could resolve sphere packing in, say, $\mathbb{R}^4$ or higher-dimensional analogues remains an active research question.