Matthew de Courcy-Ireland, Stockholm University
This talk is based on joint work with Maria Dostert and Maryna Viazovska. We prove that the Cohn–Elkies linear programming bound is not sharp for sphere packing in dimension 6. This is in contrast to Viazovska’s sharp bound in dimension 8, even though it is believed that closely related lattices achieve the optimal densities in both dimensions. The proof uses modular forms to construct feasible points in a dual program, generalizing a construction of Cohn and Triantafillou to the case of odd weight and non-trivial Dirichlet character. Non-sharpness of linear programming is demonstrated by comparing this dual bound to a stronger upper bound obtained from semidefinite programming by Cohn, de Laat, and Salmon. Our construction has vanishing Fourier coefficients along an arithmetic progression, which can be explained using skew self-adjointness of Hecke operators.