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

Step 5 of 9: Combine a

+1

and a

-1

Fourier eigenfunction into the magic function

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In plain words

It's easier to control the sign of $f$ and $\hat f$ separately if $f$ is built from pieces that are eigenfunctions of the Fourier transform — functions that come back to plus or minus themselves after transforming, so $\hat f$ is automatically known once $f$ is. Viazovska (and later, for the Leech lattice, Cohn–Kumar–Miller–Radchenko–Viazovska) build one modular-form-based eigenfunction with eigenvalue $+1$ and another with eigenvalue $-1$ , each already carrying the right double zeros, then add them together with the right relative weight to get a single function whose values at the origin satisfy $f(0)=\hat f(0)$ exactly as required.

\mathcal{F}(\varphi^{\pm}) = \pm \varphi^{\pm}, \qquad f = \varphi^{+} + \varphi^{-}

Detailed analysis

If $\mathcal{F}(\varphi) = \varphi$ (eigenvalue $+1$ ), then $\hat\varphi = \varphi$ , so any sign or zero property imposed on $\varphi$ is automatically inherited by its own transform; likewise if $\mathcal{F}(\psi) = -\psi$ then $\hat\psi = -\psi$ . Both Viazovska's $d=8$ construction and the Cohn–Kumar–Miller–Radchenko–Viazovska $d=24$ construction follow this template: build a $+1$ -eigenfunction and a $-1$ -eigenfunction, each as a contour integral of a weakly holomorphic (quasi)modular form against a kernel designed so the result has double zeros at every required lattice-shell radius, and then take a specific linear combination $f = \varphi^{+} + \varphi^{-}$ tuned so that the sign conditions $f \le 0$ outside the ball and $\hat f \ge 0$ everywhere both hold.

For the Leech lattice in $\mathbb{R}^{24}$ , the $+1$ -eigenfunction comes from a weight $-8$ , depth $2$ weakly holomorphic quasimodular form for the full modular group $\mathrm{SL}_2(\mathbb{Z})$ , and the $-1$ -eigenfunction comes from a weight $-10$ weakly holomorphic modular form for the congruence subgroup $\Gamma(2)$ ; combining them the way this template dictates is what makes the whole $d=24$ proof a direct structural echo of the $d=8$ one, rather than an unrelated new argument.

Terms in this step

Fourier eigenfunction: A function $\varphi$ that satisfies $\mathcal{F}(\varphi) = \lambda\varphi$ for some constant $\lambda$ (here $\lambda = \pm 1$ ), so its own Fourier transform is just a scalar multiple of itself.

Knowledge used in this step

Convex and discrete geometry