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 with 24 | n is a spherical 11design. 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 12designs? 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 = P i≥0 aiq i (where q = e ) 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 | ai for all i ≥ 1; (2) p | ai for all i ≥ 1 with p i, and there exists at least one j ≥ 1 with p aj ; (3) there exists at least one j ≥ 1 with p j such that p aj . We first prove that case (1) holds if and only if (p − 1) | 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. 2000 Math. Subj. Class. Primary: 05Exx; Secondary: 05B05, 11E12, 11F11, 11F30, 11F33.