Duke Mathematical Journal

Cubic rings and the exceptional Jordan algebra

Noam D. Elkies and Benedict H. Gross

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In a previous paper [EG] we described an integral structure (J, E) on the exceptional Jordan algebra of Hermitian 3×3 matrices over the Cayley octonions. Here we use modular forms and Niemeier's classification of even unimodular lattices of rank 24 to further investigate J and the integral, even lattice J0=(ZE) in J. Specifically, we study ring embeddings of totally real cubic rings A into J which send the identity of A to E, and we give a new proof of R. Borcherds's result that J0 is characterized as a Euclidean lattice by its rank, type, discriminant, and minimal norm.

Article information

Duke Math. J., Volume 109, Number 2 (2001), 383-409.

First available in Project Euclid: 5 August 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07]
Secondary: 11F27: Theta series; Weil representation; theta correspondences 11H50: Minima of forms 17C40: Exceptional Jordan structures


Elkies, Noam D.; Gross, Benedict H. Cubic rings and the exceptional Jordan algebra. Duke Math. J. 109 (2001), no. 2, 383--409. doi:10.1215/S0012-7094-01-10924-1. https://projecteuclid.org/euclid.dmj/1091737275

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