UC Berkeley

Viazovska resolved the 8-dimensional sphere packing problem by constructing the magic function for the Cohn–Elkies linear programming bound, proving that the E8 lattice packing is the densest possible packing. Soon after, Cohn, Kumar, Miller, Radchenko, and Viazovska gave a similar proof for the 24-dimensional case, showing that the Leech lattice gives the densest possible packing. One of the main steps of the proof is to verify nonpositivity and nonnegativity of the functions and their Fourier transforms, which reduces to inequalities for certain quasimodular forms. The original proofs by Viazovska and by Cohn et al. are based on interval arithmetic and Sturm’s bound, which are numerical in nature. Later, Romik gave an alternative proof of the inequalities in dimension 8.In this thesis, we give new algebraic proofs of the inequalities for both dimensions 8 and 24. In particular, we develop a theory of positive and completely positive quasimodular forms, and study how positivity interacts with derivatives and Serre derivatives of quasimodular forms. This theory is simple but powerful enough to give short proofs of the quasimodular form inequalities. We also find that the corresponding quasimodular forms are closely related to the extremal quasimodular forms by Kaneko and Koike. Along the way, we also prove that the depth 1 extremal quasimodular forms are completely positive, i.e. all the Fourier coefficients are positive, which resolves Kaneko and Koike’s conjecture in this case. The application of the theory of positive quasimodular forms is not limited to the sphere packing problem. We also study the monotonicity of functions of the form 𝑡 𝑚𝐹(𝑖𝑡) for a quasimodular form 𝐹 and 𝑡 > 0, which gives a new proof of one of the inequalities in Cohn et al.’s work on the universal optimality of E8 and the Leech lattice, and also provides a way to construct positive quasimodular forms of higher levels. We also study higher-level analogues of extremal quasimodular forms by Sakai and Tsutsumi, focusing on the positivity and integrality of their Fourier coefficients in the case of depth 1 and level Γ0(𝑁) when 𝑁 = 2, 3, 4, and also depth 2 and level Γ0(2).Finally, we give new lower and upper bounds for Bourgain, Clozel, and Kahane’s sign uncertainty principle in certain dimensions that are multiples of 4. For the upper bounds, we prove that the Fourier eigenfunctions constructed by Feigenbaum, Grabner, and Hardin are nonnegative, which gives improved upper bounds for A+(𝑑) for dimensions 𝑑 ≤ 36000. For the lower bounds, we follow Cohn and Gonçalves’ approach and use summation formulas for radial Schwartz functions associated with extremal Eisenstein series, giving improved lower bounds for A(−1) 𝑑/4+1 (𝑑) for 𝑑 ≤ 10000.