Cubic rings and the exceptional Jordan algebra. (English) Zbl 1028.11041

Elkies, Noam D.; Gross, Benedict H.

Duke Math. J. 109, No. 2, 383-409 (2001).

In a previous paper [Int. Math. Res. Not. 1996, 665-698 (1996; Zbl 0863.11027)] the authors described an integral structure \((J,E)\) on the exceptional Jordan algebra of Hermitian \(3\times 3\) matrices over the Cayley octonions and studied the integral, even lattice \(J_0 = ({\mathbb Z}E)^{\perp}\) of rank 26 and discriminant 3. In this paper they study ring embeddings \(f: A \rightarrow J\) of totally real cubic rings \(A\) into \(J\), mapping the identity element 1 of \(A\) to the polarization \(E=f(1)\) of \(J\), and give a new proof of R. Borcherds result [The Leech lattices and other lattices, Ph. D. dissertation, Cambridge (1985), see http://www.math.berkeley.edu/\(\sim\)reb/papers] that \(J_0\) is characterized as an even integral lattice of rank 26, discriminant 3, and minimal norm 4.

Reviewer: Kanat Abdukhalikov (Almaty)

MSC:

11H06	Lattices and convex bodies (number-theoretic aspects)
11H55	Quadratic forms (reduction theory, extreme forms, etc.)
11F41	Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11H50	Minima of forms
17C40	Exceptional Jordan structures

Keywords:

exceptional Jordan algebra; Euclidean lattice; Hilbert modular form

Citations:

Zbl 0863.11027