SPHERICAL DESIGNS ATTACHED TO EXTREMAL LATTICES AND THE MODULO p PROPERTY OF FOURIER COEFFICIENTS OF EXTREMAL MODULAR FORMS

A theorem of Venkov says that each nontrivial shell of an extremal even unimodular lattice in R-n with 24 vertical bar n is a spherical 11-design. It is a difficult open question whether there exists any 12-design among them. In the first part of this paper, we consider the following problem: When do all shells of an even unimodular lattice become 12-designs? We show that this does not happen in many cases, though there are also many cases yet to be answered. In the second part of this paper, we study the modulo p property of the Fourier coefficients of the extremal modular forms f - Sigma(i >= 0) a(i)q(i) (where q - e(2 pi i tau)) of weight k with k even. We are interested in determining, for each pair consisting of k and a prime p, which of the following three (exclusive) cases holds: (1) p vertical bar a(i) for all i >= 1; (2) p vertical bar a(i) for all i >= 1 with p inverted iota i, and there exists at least one j >= 1 with p inverted iota aj; (3) there exists at least one j >= 1 with p inverted iota j such that p inverted iota aj. We first prove that case (1) holds if and only if (p - 1) vertical bar k. Then we obtain several conditions which guarantee that case (2) holds. Finally, we propose a conjecture that may characterize situations in which case (2) holds.