Abstract
We derive Jacobi's quartic identity and the Borweins' cubic identity related to Ramanujan's quadratic modular equation on theta series by lattice enumerative methods. Both identities are instrumental in recent work of the Borweins on the Arithmetic Geometric Mean. Of great use are the constructions of the root lattices D 4 and E 6 by binary and ternary codes respectively. A third identity, equally due to the Borweins is also derived in relation to the root lattice E 8.
On leave of absence from CNRS, I3S, 250 rue A. Einstein, 06560 Valbonne, France
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups Springer Verlag, Grundlehren 290, 1993.
J.M. Borwein, P.B. Bor Wein, ”A cubic counterpart of Jacobi identity and the AGM” Trans. of the AMS 323, 2 (1991) 691–701.
J.M. Borwein, P.B. Borwein, F.G. Garvan, ”Some cubic modular identities of Ramanujan” Trans. of the AMS 343, 1 (1994) 35–47.
W. Feit,” Some lattices over Q(√−3).” J. of Alg. 52 (1978) 248–263.
P. Solé, ”Generalized theta functions for lattice vector quantization” DIMACS 14 (1993) 27–32.
P. Sole,” Counting Lattice Points in Pyramids” Discrete Math. to appear.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sole, P. (1995). D4, E6, E8 and the AGM. In: Cohen, G., Giusti, M., Mora, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1995. Lecture Notes in Computer Science, vol 948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60114-7_35
Download citation
DOI: https://doi.org/10.1007/3-540-60114-7_35
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60114-2
Online ISBN: 978-3-540-49440-9
eBook Packages: Springer Book Archive