Eiichi BannaiAffiliation: Kyushu University, Japan Email: bannai@math.kyushuu.ac.jp Title Of Talk: Spherical designs and toy models for D. H. Lehmer's conjecture
Abstract:
(This talk is based on joint work with Tsuyoshi Miezaki.)
In 1947, Lehmer conjectured that the Ramanujan τfunction
τ(m)
never vanishes for all positive integers m, where the τ(m)
are
the Fourier coefficients of the cusp form Δ_{24}of
weight 12.
Lehmer verified the conjecture in 1947 for m<214928639999. In 1985,
Serre verified the conjecture up to m<10^{15}, and in 1999,
Jordan and Kelly for m< 22689242781695999, and so on.
The theory of spherical tdesigns, and in particular of those which
are
the shells of Euclidean lattices, is closely related to the theory of
modular forms, as first shown by Venkov in 1984. In particular,
Ramanujan's τfunction gives the coefficients of a weighted
theta
series of the E_{8}lattice. It is shown, by Venkov, de la Harpe, and
Pache, that τ (m)=0 is equivalent to the fact that the shell of
norm 2m of the E_{8}lattice is an 8design. So, Lehmer's conjecture
is reformulated in terms of spherical tdesign.
Lehmer's conjecture is difficult to prove, and still remains open. In
this talk, we consider toy models of Lehmer's conjecture. Namely, we
show that the mth Fourier coefficient of the weighted theta series of
the Z^{2}lattice and the A_{2}lattice does not vanish,
when
the shell of norm m of those lattices is not the empty set. In other
words, the spherical 4 (resp. 6)design does not exist among the shells
in the Z^{2}lattice (resp. A_{2}lattice).
Further toy models for certain two dimensional lattices will also be
mentioned. Also, I will discuss our recent joint work with Vladimir
Yudin
on more elementary (i.e., modular form free) approach to these toy
models.
Last update made Tue Feb 16 18:41:10 EST 2010.
