Existence of a magic function in dimension 2 for the Cohn–Elkies linear programming bound

Determine the existence of a Cohn–Elkies magic function in dimension 2, i.e., a function f: R^2 → R satisfying the Cohn–Elkies linear programming hypotheses (sign conditions for f and its Fourier transform) that would certify an optimal upper bound for sphere packing density in two dimensions, analogous to the known constructions in dimensions 8 and 24.

Background

The Cohn–Elkies linear programming method shows that the existence of a suitable radial function f and its Fourier transform with prescribed sign conditions implies an upper bound on sphere packing density. Viazovska (d=8) and Cohn–Kumar–Miller–Radchenko–Viazovska (d=24) constructed such “magic functions,” proving optimality of the E8 and Leech lattice packings, respectively.

Despite intensive progress, the analogous construction in two dimensions has resisted all attempts. The paper notes that, unlike the cases d=8 and d=24 where modular and quasimodular forms lead to the magic functions, the existence of a magic function in d=2 remains unresolved.