Cusp Form Dimensions, Lattice Uniqueness, and LP Sharpness for Sphere Packing in Dimensions 8 and 24

Abstract

The Cohn-Elkies linear programming (LP) bound for sphere packing is known to be sharp in dimensions 8 and 24 but in no other dimension above 2. We investigate why by examining three independent necessary conditions for LP sharpness, drawn from number theory, lattice theory, and conformal field theory. The first condition, dim S_{d/2}(SL_2(Z)) <= 1, bounds the freedom in theta series and rules out all d >= 48. The second, derived from Cohn and Triantafillou's dual LP obstruction via cusp forms for the congruence subgroup Gamma_0(2), explains why LP sharpness fails in dimensions 16 and 32 despite the first condition being satisfied. The third, via the Hartman-Mazac-Rastelli correspondence between LP bounds and the modular bootstrap for Narain conformal field theories, reinterprets LP sharpness as the existence of an extremal CFT. We formulate a conjecture that these three conditions are equivalent for d congruent to 0 mod 8, and observe that the Bost-Connes quantum statistical system provides a natural algebraic framework in which all three perspectives are connected through the Hecke algebra.