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Spherical designs and modular forms of the \(D_4\) lattice

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Abstract

In this paper, we study shells of the \(D_4\) lattice with a slight generalization of spherical t-designs due to Delsarte–Goethals–Seidel, namely, the spherical design of harmonic index T (spherical T-design for short) introduced by Delsarte-Seidel. We first observe that, for any positive integer m, the 2m-shell of \(D_4\) is an antipodal spherical \(\{10,4,2\}\)-design on the three dimensional sphere. We then prove that the 2-shell, which is the \(D_4\) root system, is a tight \(\{10,4,2\}\)-design, using the linear programming method. The uniqueness of the \(D_4\) root system as an antipodal spherical \(\{10,4,2\}\)-design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the \(D_4\) lattice in terms of orthogonal transformations of the \(D_4\) root system, and the uniqueness of the \(D_4\) lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the \(D_4\) lattice and non-vanishing of the Fourier coefficients of a certain newform of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed.

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Data Availability

The data that support the findings of this study are available from the corresponding author, K.T. upon reasonable request.

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Acknowledgements

This work is partially supported by JSPS KAKENHI Grant Nos. 19K03445, 20K03736, 20K14294 and 22K03402, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. The authors are grateful to Prof. Eiichi Bannai, Prof. Ken Ono, Prof. Siegfried Böcheler, Prof. Jiacheng Xia and Prof. Pieter Moree for valuable discussions, comments, suggestions, and corrections. The authors also would like to extend their appreciation to the anonymous reviewers for their valuable feedback, which significantly contributed to the clarity and coherence of this paper.

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Authors and Affiliations

  1. Department of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi, 480-1198, Japan

    Masatake Hirao & Koji Tasaka

  2. Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi, 448-8542, Japan

    Hiroshi Nozaki

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  1. Masatake Hirao
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Correspondence to Koji Tasaka.

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Hirao, M., Nozaki, H. & Tasaka, K. Spherical designs and modular forms of the \(D_4\) lattice. Res. number theory 9, 77 (2023). https://doi.org/10.1007/s40993-023-00479-1

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