Abstract
In the previous paper under the same title, we showed that the m-th Fourier coefficient of the weighted theta series of the \({\mathbb{Z}}^{2}\)-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 nonempty shells in the \({\mathbb{Z}}^{2}\)-lattice (resp. A 2-lattice). This paper is the sequel to the previous paper. We take 2-dimensional lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\), then, show that the m-th Fourier coefficient of the weighted theta series of those lattices does not vanish, when the shell of norm m of those lattices is not the empty set. Equivalently, we show that the corresponding spherical 2-design does not exist among the nonempty shells in those lattices.
The second author was supported by a JSPS Research Fellowship.
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Acknowledgements
The authors thank Masao Koike for informing us that our previous results in [3] can be interpreted in terms of the cusp forms attached to L-functions with a Hecke character of CM fields, i.e., the imaginary quadratic fields \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\), and in particular for bringing our attention to Theorem 1.31 in [15]. The authors also thank Junichi Shigezumi for his helpful discussions and computations on this research.
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Bannai, E., Miezaki, T. (2013). Toy Models for D. H. Lehmer’s Conjecture II. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P. (eds) Quadratic and Higher Degree Forms. Developments in Mathematics, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7488-3_1
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