分野別セミナー

Let $\Delta_{24}(z)=q^{2}\prod_{m\geq 1}(1-q^{2m})=\sum_{m\geq 1}\tau(m)q^{2m}$
with $q=e^{\pi iz}$ be the modular form of weight 12.
Lehmer's conjecture (1947) says that $\tau(m)\not=0$ for any positive
integer $m,$ where the values of the Ramanujan function $\tau(m)$ are the
Fourier coefficients of $\Delta_{24}(z).$
Let $L$ be the $E_8$-lattice in $R^8.$ It is known by Venkov that
all the shells $L_{2m}=\{x\in L\mid ||x||^2=2m$ of $L$ are spherical
7-designs. It is known by Venkov-de la Harpe-Pache that $\tau(m)=0$
if and only if the shell $L_{2m}$ is a spherical 8-design. So, Lehmer's
conjecture is reformulated in terms of spherical designs.
It is still too difficult to answer whether there is any shell
$L_{2m}$ which is an 8-design (hence to answer Lehmer's conjecture).
Here, we answer some similar type of questions for some lattices in $R^2$,
i.e., for toy models. Namely, we obtained the following two results.
(1) All the shells $(Z^2)_m$ of the $Z^2$-lattice are spherical
3-designs, but non of them can be a 4-design.
(2) All the shells $(A_2)_{2m}$ of the $A_2$-lattice are spherical
5-designs, but non of them can be a 6-design.
We want to discuss our (so far unsuccessful) attempt to try to get similar
results for some other lattices.
(This is joint work with Tsuyoshi Miezaki of Kyushu University.)