OCTONIONS, CUBES, EMBEDDINGSMARTIN H. WEISSMAN
MARCH 2, 2009

Octonions
Let k be a field, with char (k) 6= 2. A Hurwitz algebra over k is a
finite-dimensional, unital, k-algebra A, together with a quadratic
form N : A ! k, such that the associated bilinear form is nondegenerate and:N ( xy) = N ( x ) N (y), for all x, y 2 A.
Every Hurwitz algebra over k has dimension 1, 2, 4, or 8, as a kvector space. An 8-dimensional Hurwitz k-algebra is often called aCayley or octonion algebra.
Constructing all Cayley algebras (up to isomorphism)
over Q (or a local field) is not difficult. One construction is theCayley-Dickson process: begin with a quaternion algebra B overQ. Define O = B B as a Q-vector space. Define a Q-algebra
structure on O by:(u, v) · (z, w) = (uzw̄v, wu + vz̄).

Define the main involution on O by (u, v) = (ū,v ).

{0}.

Define O to be the Q-vector space whose basis is the set {1} [P2 (F2 ). If ~x 2 P2 (F2 ), write e~x for the associated basis element of
O. Define a function f : P2 (F2 ) ⇥ P2 (F2 ) ! F2 by:f (~x, ~y) =Â x s (1) y s (1) + x s (1) y s (2) + y s (1) x s (2) x s (3) .

s 2 A3

Define a Q-algebra structure on O by identifying 1 as the unit,
and defining:(e~x ) · (e~y ) = e~x+~y · ( 1) f (~x,~y) .
The main involution on O is determined by 1̄ = 1 and e~x =Tr ( a) = N ( a + 1)
and involution ā = Tr ( a)algebras are quadratic:a2

N ( a)

N (1),a. Hurwitz

Tr ( a) + N ( a) = 0.

The isomorphism class of a Cayley k-algebra
A is determined by the isomorphism classof the quadratic space ( A, N ). The quadratic
form N of a Cayley k-algebra is always aPfister form. Every Pfister form of dimension
8 is hyperbolic or anisotropic. There are
no anisotropic forms over Q p of dimension
greater than 4. There are two dimension 8
Pfister forms over R. The isomorphism classof a quadratic form over Q is determinedby local invariants. There is no local-global
obstruction for Cayley algebras.
If b 2 B, then b̄ = Tr (b)involution.

A common construction of a “nonsplit” octonion algebra
over Q arises as follows: Let P2 (F2 ) denote the projective space oflines in F32 (a set with seven elements). Since F2 has one nonzero
element, there is a natural bijection
P2 (F2 ) $ F32

Resulting structures on a Hurwitz algebra A
include a trace,b denotes the main

Status update: The Cayley-Dickson construction is implemented in SAGE by J. Hanke andM. Weissman.
Nonsplit here means that the quadratic form
is anisotropic
Elements of P2 (F2 ) are often called
e1 , e2 , e3 , e4 , e5 , e6 , e7 in the literature. This
construction of an octonion algebra is due toFreudenthal.

Given a long history of mistakes is thesubject, the signs given by f (~x, ~y) should bemachine-verified: one should verify that one
gets an alternative algebra from this choice ofsigns, by considering quantities like (e~x e~y )e~x ,which should equal e~x (e~y e~x ).
001

e~x .

1010111111100110010

Figure 1: The Fano plane. The label abc refers
to the line through ( a, b, c) 2 P2 (F2 ).

octonions, cubes, embeddings

2Bhargava Cubes
The results on this page are all due to Manjul Bhargava1 Define a
Bhargava cube to be an element C 2 Z2 ⌦Z Z2 ⌦Z Z2 .
The group SL2 (Z )3 acts on the Z-module of Bhargava cubes,
via the tensor cube of the standard action of SL2 (Z ) on Z2 . Thereis a unique (up to normalization) quartic polynomial invariant for
the resulting geometric action; its (normalized) value is called thediscriminant, and denoted D(C ).
From a Bhargava cube C, one may construct three ordered pairsof matrices by slicing:

1Manjul Bhargava, Higher composition
laws. I. A new view on Gauss composition,and quadratic generalizations, in Annals ofMathematics, 2004.
feab

C : ( M1 , N1 ) or ( M2 , N2 ) or ( M3 , N3 ).
These yields three integer-valued quadratic forms, all of the same
discriminant D(C ):Qi ( x, y) =det( Mi xNi y), for i = 1, 2, 3.

The cube C is called projective if the three associated quadraticforms Q1 , Q2 , Q3 are primitive. Projective cubes can be brought,
by the action of SL2 (Z )3 , into a “normal form” as displayed in themargin.
The following theorem is the starting point for Bhargava’s work:
Theorem 1 (Bhargava) Fix an integer D. The orbits of SL2 (Z )3 on
the set of projective Bhargava cubes of discriminant D are in one-to-one
correspondence with the set of triples ([ Q1 ], [ Q2 ], [ Q3 ]) of SL2 (Z )equivalence classes of primitive binary quadratic forms of discriminantD, such that [ Q1 ] [ Q2 ] [ Q3 ] = [1], in the sense of Gauss composition.

ghcdFigure 2: A Bhargava cube. Slicing along thelight line yields the two matrices:
M1 =

✓acbd

◆

N1 =

✓egfh

◆

Q2 ( x, y) = 3x222xy + 2y .

.

g001mf

The significance of Bhargava’s theorem is that it provides acompletely new definition of Gauss composition. Indeed, in anygroup G, knowledge of the set of triples ( g1 , g2 , g3 ) such thatg1 g2 g3 = 1 suffices to determine the group law. In practice,
it implies that, given two primitive quadratic forms Q1 , Q2 , there
exists a projective Bhargava cube C in normal form, from which
Q3 can be easily computed.
For example, consider the following two quadratic forms ofdiscriminant 20:
Q1 ( x, y) = x2 + 5y2 .

,e0Figure 3: A Bhargava cube in normal form.
The three resulting quadratic forms, all ofdiscriminant D = m2 + 4e f g, are:Q1 ( x, y) =ex2 + mxy + f gy2 ,Q2 ( x, y) =f x2 + mxy + egy2 ,Q3 ( x, y) =gx2 + mxy + e f y2 .

200

1These fit into the cube at the right. The third quadratic form is:
Q3 ( x, y) = 2x22xy + 3y2 .

Since [ Q3 ] = [ Q2 ] and [ Q1 ] = [1], [ Q2 ] · [ Q2 ] = [1].

3021Figure 4: A Bhargava cube of discriminant
20.

octonions, cubes, embeddings

3Orders and an Embedding Question
Let O denote the nonsplit octonion algebra over Q, with basis{1} [ P2 (F2 ). We are interested in writing
down a maximal orderLin O. The naive guess Wng = Z~x 2P2 (F2 ) Ze~x at a maximalorder in O is incorrect! The correct maximal order was identified
by Coxeter2 , after many mistaken attempts (e.g. by Kirmse). Thereexists a maximal order W, which contains Wng with index 16;
Coxeter describes this order explicitly.
The following statements characterize the order W as a latticecontaining Wng : if ~x + ~y = ~z, in P2 (F2 ), then the octonion 1/2 ·(±1 ± e~x ± e~y ± e~z ) is in W. Furthermore, W contains an element~ 2 P2 (F2 ), such thatw = 1/2 · (e~x + e~y + e~z + ew
x, ~y, ~z, w~ ), for some ~N (w ) = 1.
The maximal order W (unique up to conjugation by Aut(O/k))is isometric to the E8 root lattice. The theta function of the E8 root
lattice is just the Eisenstein series E4 (of weight 4, level 1); for thisreason, there is a formula for the number of integral octonions(elements of W) of any given norm:

2H.S.M. Coxeter, Integral Cayley Numbers,
Duke Math J., 1946
The history of mistakes, and the fact thatour description of O differs from Coxeter’s,suggests we should recompute the correct
maximal order W, and check our work using
computer assistance. First,Wng ⇢ W ⇢ 1/2Wng .
One should check that the lattice W has thefollowing properties: the lattice, with thenorm form, is isometric to the E8 root lattice(with 240 elements of norm 1). It should also
be closed under multiplication.

#{w 2 W : N (w ) = n} = 240 · Â d3 .
d|nThere is a connection between Bhargava’s cubes and Coxeter’s integral octonions, arising from modular forms on exceptional groups. In my work on D4 modular forms3 , the following
quantities arise: Suppose that C is a projective cube, in normal
form, with D = D(C ) < 0, as pictured on the right.
Define:⇢(a, b, g) 2 W3 such thatEmb(C, W) =.
Na = e, Nb = f , Ng = g, Tr (abg) = m
This counts the number of embeddings of “QT-structures” (triplesof quadratic lattices endowed with a trilinear form), from a QTstructure coming from a Bhargava cube, into the QT-structurearising from O (with the norm quadratic form and the trace trilinear form).
Theorem 2 If C and C 0 are in the same SL2 (Z )3 -orbit, then Emb(C, W) =Emb(C 0 , W). The numbers Emb(C, W) are the Fourier coefficients (indexed by cubes) of a “theta function” for the group Spin4,4 .
The quantities Emb(C, W) should be machine-computable, especially for cubes of small (and perhaps prime) discriminant. Thesequantities should be identifiable with the Fourier coefficients of anEisenstein seris, but this has not yet been proven.

3M. Weissman, D4 Modular Forms, Amer. J.
of Math., 2006
g001mf0eFigure 5: A Bhargava cube in normal form.
Suppose that D = m2 + 4e f g < 0.
For three octonions a, b, g, it happens that
Tr ((ab)g) = Tr (a( bg)).
Call this quantity Tr (abg).

octonions, cubes, embeddings

4Computations and questions
Bhargava’s theorem, and the previous result, suggests that one
should carry out the following:• Choose a negative discriminant D.

• Choose a triple [ Q1 ][ Q2 ][ Q3 ] = [1] of classes of binary
quadratic forms of negative discriminant D.

1001• Find a cube C which “encodes” these classes by slicing.

• Compute the quantity Emb([ Q1 ], [ Q2 ], [ Q3 ], W) = Emb(C, W).

Such quantities fit into a general class of embedding problems
including the following:• The classical “representation of quadratic forms by quadraticforms” problem can be rephrased as counting embeddingsof a definite quadratic lattice into another definite quadraticlattice.
• If K is a quadratic imaginary field, and A is a maximal orderin K, and e is the Dirichlet character associated to K/Q, thenEmb( A, W) (the number of unital algebra embeddings) isequal to 252L(e, 2) by a result of Gross-Elkies4 .
In particular, I would hypothesize that Emb(C, W) can be computed via an Euler product. The specific form of this productshould be “guessable” from enough computations.
Observe that Emb([ Q1 ], [ Q2 ], [ Q3 ], W) assigns an integer to each“collinear” triple in a class group. There are some immediatequestions:• Do these integers depend the elements in the class group?
Do the integers only depend upon the genus of the quadratic
forms, or the spinor genus?
• Do these integers “look different” (bigger or smaller), when
the structure of the class group is different (e.g., cyclic oforder 4 or the Klein 4-group)? Can one see differences in
the fine structure of the class group from these integers (a laCohen-Lenstra)?
• Can one guess a simple Euler product, or less simple but
well-known L-value for the numbers Emb(C, W)? The degrees of the Euler factors should be guessable by computing
dimensions of Lie algebras over finite fields.
Any such answers could be helpful. Since these integers are thecoefficients of a theta function on Spin4,4 , analytic techniques areavailable to estimate (bound) these integers as the discriminant
grows.

0105

Figure 6: A Bhargava cube in normal form,
with D = 20. It reflects the identity[1] · [1] · [1] = [1]20013021Figure 7: A Bhargava cube in normal form,
with D = 20. It reflects an identity[ Q ] · [ Q ] = [1]4B. Gross and N. Elkies, Embeddings into
the integral octonions, Pac. J. of Math., 1997
Suggested computational project, involving
Emb(C, W) for the two cubes above: Count
triples (a, b, g) 2 O3 such that Na = 1,Nb = 2, Ng = 3, Tr (abg) = 2. Then,count triples (a, b, g) 2 O3 such that Na = 1,Nb = 1, Ng = 5, and Tr (abg) = 0. One
can definitely use tricks to speed up the bruteforce! Do these counts yield the same result?
What are the results?

The fact that these integers are given by
an Euler product is probably not difficult
to prove. But, there might be some very
bad Euler factors, arising from masses ofnonmaximal lattices in a rank 3 Hermitianspace.