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 $\mathbb R^n$ with $24\mid n$ 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 $f=\sum_{i\ge 0}a_iq^i$ (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) $p\mid a_i$ for all $i\ge 1$ (2) $p\mid a_i$ for all $i\ge 1$ with $p\nmid i$, and there exists at least one $j\ge 1$ with $p\nmid a_j$ (3) there exists at least one $j\ge 1$ with $p\nmid j$ such that $p\nmid a_j$. We first prove that case (1) holds if and only if $(p-1)\mid k$. 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.