Source: J. Math. Soc. Japan Volume 62, Number 3
(2010), 687-705.
In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) never vanishes for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form Δ24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 1015, and in 1999, Jordan and Kelly for m < 22689242781695999.
The theory of spherical t-design, and in particular 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 paper, 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 5 (resp. 7)-design does not exist among the shells in the Z2-lattice (resp. A2-lattice).
References
C. Bachoc and B. Venkov, Modular forms, lattices and spherical designs, Réseaux euclidiens, designs sphériques et formes modulaires, 87–111, Monogr. Enseign. Math., 37, Enseignement Math., Geneva, 2001.
E. Bannai, M. Koike, M. Shinohara and M. Tagami, Spherical designs attached to extremal lattices and the modulo $p$ property of Fourier coefficients of extremal modular forms. Mosc. Math. J., 6 (2006), 225–264.
J. H. Conway and N. J. A. Sloane, Sphere Packings Lattices and Groups, Third edition, Springer-Verlag, 1999.
P. de la Harpe and C. Pache, Cubature formulas, geometrical designs, reproducing kernels, and Markov operators, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., 248, Birkhäuser, Basel, 2005, pp.,219–267.
P. de la Harpe, C. Pache and B. Venkov, Construction of spherical cubature formulas using lattices, Algebra i Analiz, 18 (2006), 162–186; translation in St. Petersburg Math. J., 18 (2007), 119–139.
P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363–388.
Mathematical Reviews (MathSciNet):
MR485471
E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1983.
Mathematical Reviews (MathSciNet):
MR749754
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics, 298 (2005), 205–211.
N. Katz, An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), Amer. Math. Soc., Providence, R.I., 1976, pp.,275–305.
Mathematical Reviews (MathSciNet):
MR424822
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984.
Mathematical Reviews (MathSciNet):
MR766911
D. H. Lehmer, The vanishing of Ramanujan's $\tau (n)$, Duke Math. J., 14 (1947), 429–433.
Mathematical Reviews (MathSciNet):
MR21027
C. Pache, Shells of selfdual lattices viewed as spherical designs, Internat. J. Algebra Comput., 15 (2005), 1085–1127.
H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory, 54 (1995), 190–202.
B. Schoeneberg, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen (German), Math. Ann., 116 (1939), 511–523.
B. Schoeneberg, Elliptic modular functions: an introduction, Translated from the German by J. R. Smart and E. A. Schwandt, Die Grundlehren der mathematischen Wissenschaften, Band 203, Springer-Verlag, 1974.
Mathematical Reviews (MathSciNet):
MR412107
J.-P. Serre, A course in arithmetic, Translated from the French, Graduate Texts in Mathematics, No.,7. Springer-Verlag, 1973.
Mathematical Reviews (MathSciNet):
MR344216
W. Stein, Modular forms, a computational approach, With an appendix by Paul E. Gunnells, Graduate Studies in Mathematics, 79, American Mathematical Society, Providence, RI, 2007.
J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), Lecture Notes in Math., 1240, Springer, 1987, pp.,275–280.
Mathematical Reviews (MathSciNet):
MR894516
B. B. Venkov, Even unimodular extremal lattices (Russian), Algebraic geometry and its applications, Trudy Mat. Inst. Steklov., 165 (1984), 43–48; translation in Proc. Steklov Inst. Math., 165 (1985), 47–52.
Mathematical Reviews (MathSciNet):
MR752931
B. B. Venkov, Réseaux et designs sphériques (French), [Lattices and spherical designs] Réseaux euclidiens, designs sphériques et formes modulaires, 10–86, Monogr. Enseign. Math., 37, Enseignement Math., Geneva, 2001.
