Shells of selfdual lattices viewed as spherical designs

International Journal of Algebra and Computation

Volumn 15, Issue 5-6, 2005, Pages 1085-1127

  Pache, Claude ^a  

^a UNIVERSITY OF GENEVA   (Switzerland)

Author keywords

Modular form; Selfdual lattice; Spherical design

Indexed keywords

EID: 30344482515     PISSN: 02181967     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218196705002797     Document Type: Conference Paper

References (30)

1. n, J. Théor. Nombres Bordeaux 9(2) (1997) 479-497.
2. Ch. Bachoc and B. B. Venkov, Modular forms, lattices and spherical designs, [16, Chap. 2].
3. B. Bekka and P. de la Harpe, Irreducibility of unitary group representations and reproducing kernels Hubert spaces, Expo. Math. 21(2) (2003) 115-149.
4. R. E. Borcherds, The Leech lattice and other latices, Ph.D. dissertation, University of Cambridge (1984).
5. R. E. Borcherds, The 24-dimensional odd unimodular lattices [7, Chap. 17].
6. N. Bourbaki, Groupes et Algèbres de Lie (Masson, Paris, 1981).
7. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edn., (Springer, New York, 1999).
8. P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977) 363-388.
9. W. Ebeling, Lattices and Codes, A Course Partially Based on Lectures by F. Hirzenbruch (Vieweg, Braunschweig, 1994). 2nd revised edn., 2002.
10. n-lattice, Math. Res. Lett. 2(3) (1995) 321-326.
11. N. D. Elkies, Lattices and codes with long shadows, Math. Res. Lett. 2(5) (1995) 643-645.
12. J.-M. Goethals and J. J. Seidel, Spherical designs, Proc. Sympos. Pure Math. XXXIV (Amer. Math. Soc., Providence R.I. 1979), pp. 255-272.
13. J.-M. Goethals and J. J. Seidel, Cubature formulae, polytopes, and spherical designs, in The Geometric Vein. The Coxeter Festschrift (Springer, New York, Berlin, 1981), pp. 203-218.
14. P. de la Harpe and C. Pache, Spherical designs and finite group representations (some results of E. Bannai), European J. Combin. 25(2) (2004) 213-227.
15. P. de la Harpe and C. Pache, Cubature formulas, geometrical designs, reproducing kernels, and Markov operators, in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math. 248 (Birkhäuser, 2005), pp. 219-268).
16. J. Martinet (ed.), Réseaux euclidiens, designs sphériques et formes modulaires, Autour des travaux de B. Venkov, L'Enseignement Mathématique, Monographie No. 37 (Genève, 2001).
17. G. Nebe and B. B. Venkov, The strongly perfect lattices of dimension 10, J. Théor. Nombres Bordeaux 12(2) (2000) 503-518.
18. R. A. Rankin, Modular Forms and Functions (Cambridge University Press, 1977).
19. R. Scharlau and R. Schulze-Pillot, Extremal lattices, in Algorithmic algebra and Number Theory, Heidelberg, 1997 (Springer, Berlin, 1999), pp. 139-170.
20. J.-P. Serre, Sur la lacunarité des puissances de 77, Glasg. Math. J. 27 (1985) 203-221
21. (Œuvres, Vol. IV, pp. 66-84, see also 640).
22. B. B. Venkov, The classification of integral even unimodular 24-dimensional quadratic forms, Trudy Mat. Inst. Steklova 148 (1978) 65-76,
23. Proc. Steklov Inst. Math. No. 4 (1980) 63-74 (Even unimodular 24-dimensional lattices [7, Chap. 18]).
24. B. B. Venkov, Even unimodular extremal lattices, Tr. Mat. Inst. Steklova 165 (1984) 43-48 (in Russian)
25. Proc. Steklov Inst. Math. 165(3) (1985) 47-52.
26. B. B. Venkov (notes by J. Martinet), Réseaux et designs sphériques [16, Chap. 1].
27. N. Ya. Vilenkin, Spetsial'nye funktsii i teoriya predstavleniǐ grupp, 2nd edn., (Nauka, Moscow, 1991),
28. Special Functions and the Theory of Group Representations, Transl. Mathematical Monographs, Vol. 22 (Amer. Math. Soc. Providence R.I., 1968),
29. Fonctions spéciales et théorie de la représentation des groupes (Dunod, Paris, 1969).
30. G. Voronoi, Nouvelles applications des paramètres continus àla théorie des formes quadratiques : 1. Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math. 133 (1908) 97-178.