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

Step 4 of 9: Constructing the magic function from modular forms in

d=8

Previous step Next step

In plain words

Building a function with double zeros at infinitely many precise radii ( $\sqrt2, 2, \sqrt6, \dots$ ) sounds impossible with ordinary calculus, but modular forms are functions on the upper half-plane with an enormous amount of built-in symmetry, and a century of number theory provides ready-made tools (contour integrals against them) for manufacturing exactly this kind of infinitely-constrained function; Viazovska's insight was to see that the $E_8$ packing problem was secretly a modular-forms problem in disguise.

f = (1 + \text{lower order}) \cdot \left(\text{quasimodular piece} \right), \quad d = 8,\; r = \sqrt{2}

Detailed analysis

For the $E_8$ lattice the required radius is $r = \sqrt{2}$ , matching its minimal vector length. Viazovska built the magic function $f$ as a modular integral transform (a contour integral against weakly holomorphic quasimodular forms for $\mathrm{SL}_2(\mathbb{Z})$ ), engineered so that $f$ and its Fourier transform $\hat f$ have double zeros at every nonzero $E_8$ vector length $\sqrt{2n}$ with $n\ge1$ , while modularity forces $f(0) = \hat f(0)$ .

Terms in this step

weakly holomorphic modular form: A function on the upper half-plane that transforms in a precise, highly symmetric way under a group of transformations (like $z \mapsto -1/z$ ), is holomorphic except possibly at one boundary point, and can be expanded as a power series in $q=e^{2\pi i z}$ .

Knowledge used in this step

Convex and discrete geometry