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Spherical designs and modular forms of the D4 lattice.
- Source :
-
Research in Number Theory . 11/1/2023, Vol. 9 Issue 4, p1-18. 18p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MODULAR design
*CONGRUENCE lattices
*LINEAR programming
*THETA functions
Subjects
Details
- Language :
- English
- ISSN :
- 25220160
- Volume :
- 9
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Research in Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 173395803
- Full Text :
- https://doi.org/10.1007/s40993-023-00479-1