Affiliation: Kyushu University, Japan
Title Of Talk: Spherical designs and toy models for D. H. Lehmer's conjecture
(This talk is based on joint work with Tsuyoshi Miezaki.)
In 1947, Lehmer conjectured that the Ramanujan τ-function
never vanishes for all positive integers m, where the τ(m)
the Fourier coefficients of the cusp form Δ24of
Lehmer verified the conjecture in 1947 for m<214928639999. In 1985,
Serre verified the conjecture up to m<1015, and in 1999,
Jordan and Kelly for m< 22689242781695999, and so on.
The theory of spherical t-designs, and in particular of those which
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 τ-function gives the coefficients of a weighted
series of the E8-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 E8-lattice is an 8-design. So, Lehmer's conjecture
is reformulated in terms of spherical t-design.
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 m-th Fourier coefficient of the weighted theta series of
the Z2-lattice and the A2-lattice does not vanish,
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 Z2-lattice (resp. A2-lattice).
Further toy models for certain two dimensional lattices will also be
mentioned. Also, I will discuss our recent joint work with Vladimir
on more elementary (i.e., modular form free) approach to these toy
Last update made Tue Feb 16 18:41:10 EST 2010.