Eiichi Bannai

Affiliation: Kyushu University, Japan

Email: bannai@math.kyushu-u.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 τ-function τ(m) never vanishes for all positive integers m, where the τ(m) are the Fourier coefficients of the cusp form Δ24of weight 12. 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 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 τ-function gives the coefficients of a weighted theta 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, 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 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 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.
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