Tài liệu Đề tài " Higher composition laws II: On cubic analogues of Gauss composition " ppt

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868 MANJUL BHARGAVA
We observe that the action of GL
2
(Z) × SL
3
(Z) × SL
3
(Z) on its
18-dimensional representation Z
2
⊗ Z
3
⊗ Z
3
has just a single polynomial in-
variant.
2
Indeed, the action of SL
3
(Z) × SL
3
(Z)onZ
2
⊗ Z
3
⊗ Z
3
has four
independent invariants, namely the coefficients of the binary cubic form
(1) f(x, y) = Det(Ax − By).
The group GL
2
(Z) acts on the cubic form f(x, y), and it is well-known that
this action has exactly one polynomial invariant (see, e.g., [7]), namely the dis-
criminant Disc(f)off. Hence the unique GL
2
(Z) ×SL
3
(Z) ×SL
3
(Z)-invariant
on Z
2
⊗ Z
3
⊗ Z
3
is given by Disc(Det(Ax − By)). We call this fundamental in-
variant the discriminant of (A, B), and denote it by Disc(A, B). If Disc(A, B)
is nonzero, we say that (A, B)isanondegenerate element of Z
2
⊗ Z
3
⊗ Z
3
.
Similarly, we call a binary cubic form f nondegenerate if Disc(f) is nonzero.
2.2. The parametrization of cubic rings. The parametrization of cubic or-
ders by integral binary cubic forms was first discovered by Delone and Faddeev
in their famous treatise on cubic irrationalities [4]; this parametrization was
refined recently to general cubic rings by Gan-Gross-Savin [5] and by Zagier
(unpublished). Their construction is as follows. Given a cubic ring R (i.e., any
ring free of rank 3 as a Z-module), let 1,ω,θ be a Z-basis for R. Translating
ω, θ by the appropriate elements of Z, we may assume that ω ·θ ∈ Z. We call a
basis satisfying the latter condition normalized, or simply normal.If1,ω,θ
is a normal basis, then there exist constants a, b, c, d,, m, n ∈ Z such that
(2)
ωθ = n
ω
2
= m + bω − aθ
θ
2
=  + dω − cθ.
To the cubic ring R with multiplication table as above, we associate the binary
cubic form f(x, y)=ax
3
+ bx
2
y + cxy
2
+ dy
3
.
Conversely, given a binary cubic form f(x, y)=ax
3
+ bx
2
y + cxy
2
+ dy
3
,
form a potential cubic ring having multiplication laws (2). The values of , m, n
are subject to the associative law relations ωθ · θ = ω · θ
2
and ω
2
· θ = ω · ωθ,
which when multiplied out using (2), yield a system of equations possessing
the unique solution (n, m, )=(−ad, −ac, −bd), thus giving
(3)
ωθ = −ad
ω
2
= −ac + bω − aθ
θ
2
= −bd + dω − cθ.
If follows that any binary cubic form f(x, y)=ax
3
+ bx
2
y + cxy
2
+ dy
3
, via
the recipe (3), leads to a unique cubic ring R = R(f).
2
As in [2], we use the convenient phrase “single polynomial invariant” to mean that the
polynonomial invariant ring is generated by one element.
HIGHER COMPOSITION LAWS II
869
Lastly, one observes by an explicit calculation that changing the Z-basis
ω, θ of R/Z by an element of GL
2
(Z), and then renormalizing the basis in
R, transforms the resulting binary cubic form f(x, y) by that same element
of GL
2
(Z).
3
Hence an isomorphism class of cubic ring determines a binary
cubic form uniquely up to the action of GL
2
(Z). It follows that isomorphism
classes of cubic rings are parametrized by integral binary cubic forms modulo
GL
2
(Z)-equivalence.
One finds by a further calculation that the discriminant of a cubic ring
R(f) is precisely the discriminant of the binary cubic form f. We summarize
this discussion as follows:
Theorem 1 ([4],[5]). There is a canonical bijection between the set of
GL
2
(Z)-equivalence classes of integral binary cubic forms and the set of iso-
morphism classes of cubic rings, by the association
f ↔ R(f).
Moreover, Disc(f ) = Disc(R(f)).
We say a cubic ring is nondegenerate if it has nonzero discriminant (equiv-
alently, if it is an order in an ´etale cubic algebra over Q). The discriminant
equality in Theorem 1 implies, in particular, that nondegenerate cubic rings
correspond bijectively with equivalence classes of nondegenerate integral bi-
nary cubic forms.
2.3. Cubic rings and 2 × 3 × 3 boxes of integers. In this section we
classify the nondegenerate Γ-orbits on Z
2
⊗ Z
3
⊗ Z
3
in terms of ideal classes
in cubic rings. Before stating the result, we recall some definitions. As in [2],
we say that a pair (I,I

) of (fractional) R-ideals in K = R ⊗ Q is balanced if
II

⊆ R and N(I)N(I

) = 1. Furthermore, two such balanced pairs (I
1
,I

1
)
and (I
2
,I

2
) are called equivalent if there exists an invertible element κ ∈ K
such that I
1
= κI
2
and I

1
= κ
−1
I

2
. For example, if R is a Dedekind domain
then an equivalence class of balanced pairs of ideals is simply a pair of ideal
classes that are inverse to each other in the ideal class group.
Theorem 2. There is a canonical bijection between the set of nondegen-
erate Γ-orbits on the space Z
2
⊗ Z
3
⊗ Z
3
and the set of isomorphism classes
of pairs (R, (I,I

)), where R is a nondegenerate cubic ring and (I,I

) is an
equivalence class of balanced pairs of ideals of R. Under this bijection, the
discriminant of an integer 2 × 3 × 3 box equals the discriminant of the corre-
sponding cubic ring.
3
In basis-free terms, the binary cubic form f represents the mapping R/
Z
→∧
3
R
∼
=
Z
given by ξ → 1 ∧ ξ ∧ ξ
2
, making this transformation property obvious.
870 MANJUL BHARGAVA
Proof. Given a pair of balanced R-ideals I and I

, we first show how to
construct a corresponding pair (A, B)of3× 3 integer matrices. Let 1,ω,θ
denote a normal basis of R, and let α
1
,α
2
,α
3
 and β
1
,β
2
,β
3
 denote any
Z-bases for the ideals I and I

having the same orientation as 1,ω,θ. Then
since II

⊆ R, we must have
(4)
α
i
β
j
= c
ij
+ b
ij
ω + a
ij
θ
for some set of twenty-seven integers a
ij
, b
ij
, and c
ij
, where i, j ∈{1, 2, 3}.
Let A and B denote the 3 × 3 matrices (a
ij
) and (b
ij
) respectively. Then
(A, B) ∈ Z
2
⊗ Z
3
⊗ Z
3
is our desired pair of 3 × 3 matrices.
By construction, it is clear that changing α
1
,α
2
,α
3
 or β
1
,β
2
,β
3
 to
some other basis of I or I

via a matrix in SL
3
(Z) would simply transform A
and B by left or right multiplication by that same matrix. Similarly, a change
of basis from 1,ω,θ to another normal basis 1,ω

,θ

 of R is completely
determined by a unique element

rs
uv

∈ GL
2
(Z), where
ω

= q + rω + sθ
θ

= t + uω + vθ
for some integers q,t. It is easily checked that this change of basis transforms
(A, B) by the same element

rs
uv

∈ GL
2
(Z). Conversely, any pair of 3 × 3
matrices in the same Γ-orbit as (A, B) can actually be obtained from (R, (I,I

))
in the manner described above, simply by changing the bases for R, I, and I

appropriately.
Next, suppose (J, J

) is a balanced pair of ideals of R that is equivalent
to (I,I

), and let κ be the invertible element in R ⊗ Q such that J = κI and
J

= κ
−1
I

. If we choose bases for I,I

,J,J

to take the form α
1
,α
2
,α
3
,
β
1
,β
2
,β
3
, κα
1
,κα
2
,κα
3
, and κ

β
1
,κ

β
2
,κ

β
3
 respectively, then it is im-
mediate from (4) that (R, (I,I

)) and (R, (J, J

)) will yield identical elements
(A, B)inZ
2
⊗ Z
3
⊗ Z
3
. It follows that the association (R, (I,I

)) → (A, B)is
a well-defined map even on the level of equivalence classes.
It remains to show that our mapping (R, (I,I

)) → (A, B) from the set
of equivalence classes of pairs (R, (I,I

)) to the space (Z
2
⊗ Z
3
⊗ Z
3
)/Γis
in fact a bijection. To this end, let us fix the 3 × 3 matrices A =(a
ij
) and
B =(b
ij
), and consider the system (4), which at this point consists mostly of
indeterminates. We show in several steps that these indeterminates are in fact
essentially determined by the pair (A, B).
First, we claim that the ring structure of R = 1,ω,θ is completely de-
termined. Indeed, let us write the multiplication in R in the form (3), with
unknown integers a, b, c, d, and let f = ax
3
+ bx
2
y + cxy
2
+ dy
3
. We claim that
the system of equations (4) implies the following identity:
(5) Det(Ax − By)=N(I)N(I

) · (ax
3
+ bx
2
y + cxy
2
+ dy
3
).
HIGHER COMPOSITION LAWS II
871
To prove this identity, we begin by considering the simplest case, where we
have I = I

= R, with identical Z-bases α
1
,α
2
,α
3
 = β
1
,β
2
,β
3
 = 1,ω,θ.
In this case, from the multiplication laws (3) we see that the pair (A, B)in(4)
is given by
(6) (A, B)=






1
−a
1 −c



,



1
1 b
d






.
For this (A, B), one finds that indeed Disc(Ax−By)=ax
3
+bx
2
y +cxy
2
+dy
3
,
proving the identity in this special case.
Now suppose that I and I

are changed to general fractional ideals of
R, having Z-bases α
1
,α
2
,α
3
 and β
1
,β
2
,β
3
 respectively. Then there exist
transformations T , T

∈ SL
3
(Q) taking 1,ω,θ to the new bases α
1
,α
2
,α
3

and β
1
,β
2
,β
3
 respectively, and so the new (A, B) in (4) may be obtained by
transforming the pair of matrices on the right side of (6) by left multiplication
by T and by right multiplication by T

. The binary cubic form Det(Ax−By)is
therefore seen to multiply by a factor of det(T ) det(T

)=N(I)N(I

), proving
identity (5) for general I and I

.
Now by assumption we have N(I)N(I

) = 1, so identity (5) implies
(7) Det(Ax − By)=f(x, y)=ax
3
+ bx
2
y + cxy
2
+ dy
3
;
thus the matrices A and B do indeed determine f(x, y) and hence the ring R.
Next, we show that the quantities c
ij
in (4) are also completely determined
by A and B. By the associative law in R, we have nine equations of the form
(8) (α
i
β
j
)(α
i

β
j

)=(α
i
β
j

)(α
i

β
j
),
for 1 ≤ i, i

,j,j

≤ 3. Expanding these identities out using (4), (3), and (7),
and then equating the coefficients of 1, ω, and θ, yields a system of 18 linear
and 9 quadratic equations in the 9 indeterminates c
ij
in terms of a
ij
and b
ij
.
We find that this system has exactly one (quite pretty) solution, given by
(9) c
ij
=

i

<i

,j

<j


ii

i

123

jj

j

123





a
ij
a
ij

a
i

j
a
i

j





·




b
ij
b
ij

b
i

j
b
i

j





where

rst
123

denotes the sign of the permutation (r, s, t)of(1, 2, 3). (Note that
the solutions for the {c
ij
} are necessarily integral, since they are polynomials
in the a
ij
and b
ij
!) Thus the c
ij
’s are also uniquely determined by (A, B).
We still must determine the existence of α
i
,β
j
∈ R yielding the desired
a
ij
, b
ij
, and c
ij
’s in (4). An examination of the system (4) shows that we have
(10) α
1
: α
2
: α
3
= c
1j
+b
1j
ω +a
1j
θ : c
2j
+b
2j
ω +a
2j
θ : c
3j
+b
3j
ω +a
3j
θ,
for any 1 ≤ j ≤ 3. That the ratio on the right-hand side of (10) is independent
of the choice of j follows from the identities (8) that we have forced on the
872 MANJUL BHARGAVA
system (4). Thus the triple (α
1
,α
2
,α
3
) is uniquely determined up to a factor
in K
∗
. Once the basis α
1
,α
2
,α
3
 of I is chosen, then the basis β
1
,β
2
,β
3
 for
I

is given directly from (4), since the c
ij
, b
ij
, and a
ij
are known. Therefore
the pair (I,I

) is uniquely determined up to equivalence.
To see that this object (R, (I,I

)) as determined above forms a valid pair in
the sense of Theorem 2, we must only check that I and I

, currently given only
as Z-modules in K, are actually fractional ideals of R. In fact, using explicit
embeddings of I and I

into K, or by examining (4) directly, one can calculate
the exact R-module structures of I

and I explicitly in terms of (A, B); these
module structures are too beautiful to be left unmentioned.
Given a matrix M, let us use M
i
to denote the i-th column of M and |M|
to denote the determinant of M. Then the R-module structure of I

is given
by
(11)
−ω · α
1
= |B
1
A
2
A
3
|·α
1
+ |A
1
B
1
A
3
|·α
2
+ |A
1
A
2
B
1
|·α
3
−ω · α
2
= |B
2
A
2
A
3
|·α
1
+ |A
1
B
2
A
3
|·α
2
+ |A
1
A
2
B
2
|·α
3
−ω · α
3
= |B
3
A
2
A
3
|·α
1
+ |A
1
B
3
A
3
|·α
2
+ |A
1
A
2
B
3
|·α
3
−θ · α
1
= |A
1
B
2
B
3
|·α
1
+ |B
1
A
1
B
3
|·α
2
+ |B
1
B
2
A
1
|·α
3
−θ · α
2
= |A
2
B
2
B
3
|·α
1
+ |B
1
A
2
B
3
|·α
2
+ |B
1
B
2
A
2
|·α
3
−θ · α
3
= |A
3
B
2
B
3
|·α
1
+ |B
1
A
3
B
3
|·α
2
+ |B
1
B
2
A
3
|·α
3
,
while the R-module structure of I is given analogously in terms of the rows of A
and B rather than the columns. It is evident that all the structure coefficients
above are integers, and this concludes the proof of Theorem 2.
Our discussion makes the bijection of Theorem 2 very precise. Given a
cubic order R and a balanced pair (I,I

) of ideals in R, the corresponding
element (A, B) ∈ Z
2
⊗ Z
3
⊗ Z
3
is obtained from the set of equations (4).
Conversely, given an element (A, B) ∈ Z
2
⊗ Z
3
⊗ Z
3
, the ring R is determined
by (3) and (7); bases for the ideal classes I and I

of R may be obtained from
(10) and (4), and the R-module structures of I and I

are given by (11).
Note that the algebraic formulae in the proof of Theorem 2 could be
used to extend the bijection also to degenerate orbits, i.e., orbits where the
discriminant is zero. Such orbits correspond to cubic rings R of discriminant
zero, together with a balanced pair of R-modules I,I

having rank 3 over Z.
The condition of “balanced”, however, becomes even harder to understand in
the degenerate case! To avoid such technicalities we have stated the result only
in the primary cases of interest, namely those involving nondegenerate orbits
and rings.
4
4
It is an interesting question to formulate a module-theoretic definition of “balanced”
that applies over any ring, and that is functorial (i.e., respects extension by scalars). This
would allow one to directly extend Theorem 2 both to degenerate orbits and to orbits over
an arbitrary commutative ring.
HIGHER COMPOSITION LAWS II
873
The proof of Theorem 2 not only gives a complete description of the
nondegenerate orbits of the representation of Γ on Z
2
⊗ Z
3
⊗ Z
3
in terms of
cubic rings, but also allows us to precisely determine the point stabilizers. We
have the following
Corollary 3. The stabilizer in Γ of a nondegenerate element (A, B) ∈
Z
2
⊗ Z
3
⊗ Z
3
is given by the semidirect product
Aut(R)  U
+
(R
0
),
where (R, (I,I

)) is the pair corresponding to (A, B) as in Theorem 2, R
0
=
End
R
(I) ∩ End
R
(I

) is the intersection of the endomorphism rings of I and I

,
and U
+
(R
0
) denotes the group of units of R
0
having positive norm.
Note that if I,I

are projective R-modules, then R
0
= R, so that the
stabilizer of (A, B) in Γ is simply Aut(R)  U
+
(R). This is in complete anal-
ogy with Gauss’s case of binary quadratic forms, where generic stablizers are
given by the groups of units of positive norm in the corresponding quadratic
endomorphism rings.
Proof. The proof of Theorem 2 shows that an element (A, B) uniquely
determines the multiplication table of R, in terms of some basis 1,ω,θ. Ele-
ments of GL
2
(Z) that send this basis to another basis 1,ω

,θ

 with the iden-
tical multiplication table evidently correspond to elements of Aut(R). Once
this automorphism has been fixed, the system of equations (10) then uniquely
determines the triples (α
1
,α
2
,α
3
) and (β
1
,β
2
,β
3
) up to factors κ, κ
−1
∈ K
∗
.
It follows that an element T × T

∈ SL
3
(Z) × SL
3
(Z) acting on the bases
α
1
,α
2
,α
3
 and β
1
,β
2
,β
3
 of I and I

respectively will preserve (10) if and
only if Tα
i
= κα
i
and T

β
j
= κ
−1
β
j
. In other words, T acts as multiplica-
tion by a unit κ in the endomorphism ring of I, while T

acts as the inverse
κ
−1
∈ End
R
(I

)onI

. This is the desired conclusion.
2.4. Cubic rings and pairs of ternary quadratic forms. Just as we were
able to impose a symmetry condition on 2×2×2 matrices to obtain information
on the exponent 3-parts of class groups of quadratic rings ([2, §2.4]), we can
impose a symmetry condition on 2 × 3 × 3 matrices to yield information on
the exponent 2-parts of class groups of cubic rings. The “symmetric” elements
in Z
2
⊗ Z
3
⊗ Z
3
are precisely the elements of Z
2
⊗ Sym
2
Z
3
, i.e., pairs (A, B)
of symmetric 3 × 3 integer matrices, or equivalently, pairs (A, B) of integral
ternary quadratic forms. The cubic form invariant f and the discriminant
Disc(A, B)of(A, B) may be defined in the identical manner; we have f(x, y)=
Det(Ax−By) and Disc(A, B) = Disc(Det(Ax−By)). Again, we say an element
(A, B) ∈ Z
2
⊗ Sym
2
Z
3
is nondegenerate if Disc(A, B) is nonzero.
874 MANJUL BHARGAVA
The precise correspondence between nondegenerate pairs of ternary
quadratic forms and ideal classes “of order 2” in cubic rings is then given
by the following theorem.
Theorem 4. There is a canonical bijection between the set of nondegen-
erate GL
2
(Z) × SL
3
(Z)-orbits on the space Z
2
⊗ Sym
2
Z
3
and the set of equiv-
alence classes of triples (R, I, δ), where R is a nondegenerate cubic ring, I is
an ideal of R, and δ is an invertible element of R ⊗ Q such that I
2
⊆ (δ)
and N(δ)=N(I)
2
. (Here two triples (R, I, δ) and (R

,I

,δ

) are equivalent if
there exists an isomorphism φ : R → R

and an element κ ∈ R

⊗ Q such
that I

= κφ(I) and δ

= κ
2
φ(δ).) Under this bijection, the discriminant of
a pair of ternary quadratic forms equals the discriminant of the corresponding
cubic ring.
Proof. For a triple (R, I, δ) as above, we first show how to construct a
corresponding pair (A, B) of ternary quadratic forms. Let 1,ω,θ denote a
normal basis of R, and let α
1
,α
2
,α
3
 denote a Z-basis of the ideal I having
the same orientation as 1,ω,θ. Since by hypothesis I is an ideal whose square
is contained in δ · R, we must have
(12) α
i
α
j
= δ ( c
ij
+ b
ij
ω + a
ij
θ )
for some set of integers a
ij
, b
ij
, and c
ij
. Let A and B denote the 3 × 3
symmetric matrices (a
ij
) and (b
ij
) respectively. Then the ordered pair (A, B) ∈
Z
2
⊗ Sym
2
Z
3
is our desired pair of ternary quadratic forms.
The matrices A and B can naturally be viewed as quadratic forms on
the lattice I = Zα
1
+ Zα
2
+ Zα
3
. Hence changing α
1
,α
2
,α
3
 to some other
basis of I, via an element of SL
3
(Z), would simply transform (A, B) (via the
natural SL
3
(Z)-action) by that same element. Also, just as in Theorem 2, a
change of the basis 1,ω,θ to another normal basis by an element of GL
2
(Z)
transforms (A, B) by that same element. We conclude that our map from
equivalence classes of triples (R, I, δ) to equivalence classes of pairs (A, B)of
ternary quadratic forms is well-defined.
To show that this map is a bijection, we fix the pair A =(a
ij
) and B =
(b
ij
) of ternary quadratic forms, and then show that these values determine
all the indeterminates in the system (12). First, to show that the ring R is
determined, we assume that R has multiplication given by the equations in
(3) for unknown integers a, b, c, d, and as in the proof of Theorem 2, we derive
from (12) the identity
(13)
Det(Ax − By)=N(I)
2
N(δ)
−1
· (ax
3
+ bx
2
y + cxy
2
+ dy
3
)
= ax
3
+ bx
2
y + cxy
2
+ dy
3
,
where we have used the hypothesis that N (δ)=N(I)
2
. It follows as before
that the ring R is determined by the pair (A, B).
HIGHER COMPOSITION LAWS II
875
Next we use the associative law in R to show that the constants c
ij
in
the system (12) are uniquely determined. We have three identities of the form
(δ
−1
α
2
1
)(δ
−1
α
2
2
)=(δ
−1
α
1
α
2
)
2
, and three more of the form (δ
−1
α
2
1
)(δ
−1
α
2
α
3
)=
(δ
−1
α
1
α
2
)(δ
−1
α
1
α
3
). Expanding out all six of these using (3) and (12), and
then equating the coefficients of 1, ω, and θ, yields a system of 18 linear and
quadratic equations in the six indeterminates c
11
,c
22
,c
33
,c
12
,c
13
,c
23
. This
system in the c
ij
has a unique solution, given again by (9).
Now an examination of the system (12) shows that we have
(14) α
1
: α
2
: α
3
= c
1j
+ b
1j
ω +a
1j
θ : c
2j
+ b
2j
ω +a
2j
θ : c
3j
+ b
3j
ω +a
3j
θ,
and the latter ratio is independent of the choice of j ∈{1, 2, 3}. Thus the
triple (α
1
,α
2
,α
3
) is uniquely determined up to a factor in R. Regardless of
how the triple (α
1
,α
2
,α
3
) is scaled, this then determines δ uniquely up to a
square factor in R.
Finally, to see that this object (R, I, δ) is really a valid triple in the sense
of Theorem 4, we must only check that I is an ideal of R. Again, the R-module
structure of I can be determined explicitly in terms of (A, B), and is given by
(11). This completes the proof of Theorem 4.
The proof gives very precise information about the bijection of Theorem 4.
Given a triple (R, I, δ), the corresponding pair (A, B) of ternary quadratic
forms is obtained from equations (12). Conversely, given an element (A, B) ∈
Z
2
⊗ Sym
2
Z
3
, the ring R is determined by (3) and (13); a basis for the ideal
class I may be obtained from (14), and the R-module structure of I is given
by (11).
Again, we may determine precisely the point stabilizers:
Corollary 5. The stabilizer in GL
2
(Z) × SL
3
(Z) of a nondegenerate
element (A, B) ∈ Z
2
⊗ Sym
2
Z
3
is given by the semidirect product
Aut(R)  U
+
2
(R
0
),
where (R, I) is the pair corresponding to (A, B) as in Theorem 4, R
0
= End
R
(I)
is the endomorphism ring of I, and U
+
2
(R
0
) denotes the group of units of R
0
having order dividing 2 and positive norm.
Note that Aut(R) is contained in the symmetric group S
3
, while U
+
2
(R)
must be contained in the Klein-four group K
4
. It follows that the stabilizers
occurring in Corollary 5 are contained in the finite group S
3
 K
4
= S
4
. This
is consistent with the results of Sato-Kimura [9] and Wright-Yukie [12] over
fields.
If I is projective over R, then R
0
= R so that the stabilizer of (A, B)is
simply given by Aut(R)  U
+
2
(R). Corollary 5 may be proven in a manner
similar to Corollary 3, and so we omit the proof.
876 MANJUL BHARGAVA
2.5. Cubic rings and pairs of senary alternating 2-forms. As in [2], rather
than a symmetry condition we may impose instead a skew-symmetry condition
on Z
2
⊗ Z
3
⊗ Z
3
using the “fusion” operator of [2, Section 2.6]. More precisely,
if we realize elements of the space Z
2
⊗∧
2
Z
6
as pairs of skew-symmetric 6 × 6
matrices (A, B), then there is a natural map
(15) id ⊗∧
3,3
: Z
2
⊗ Z
3
⊗ Z
3
→ Z
2
⊗∧
2
Z
6
defined by sending
(16) (A, B) →

A
−A
t

,

B
−B
t

.
The resulting skew-symmetrized space Z
2
⊗∧
2
Z
6
has a natural action by the
group GL
2
(Z) × SL
6
(Z), and this group action again possesses a unique poly-
nomial invariant. Indeed, a complete set of invariants for the action of SL
6
(Z)
on Z
2
⊗∧
2
Z
6
is given by the four coefficients of the binary cubic form
f(x, y) = Pfaff(Ax −By),
and so the unique GL
2
(Z)×SL
6
(Z)-invariant is given by Disc(Pfaff(Ax−By)),
which we again call the discriminant Disc(A, B)of(A, B). It is evident from the
explicit formula (16) that the map (15) is discriminant-preserving. As usual,
we say an element in Z
2
⊗∧
2
Z
6
is nondegenerate if it has nonzero discriminant.
Consistent with the pattern laid down in [2], the fused space Z
2
⊗∧
2
Z
6
leads to the parametrization of certain rank 2 modules over cubic rings. Sup-
pose R is any nondegenerate cubic ring, and let K = R ⊗ Q. As in [2], we
consider rank 2 modules M over R as equipped with explicit embeddings into
K ⊕ K, i.e., as rank 2 ideals. Moreover, we say a rank 2 ideal M ⊆ K ⊕ K is
balanced if Det(M ) ⊆ R and N(M) = 1. Finally, two such rank 2 ideals are
equivalent if one can be mapped to the other via an element of SL
2
(K). Our
parametrization result is then as follows:
Theorem 6. There is a canonical bijection between the set of nondegen-
erate GL
2
(Z) × SL
6
(Z)-orbits on the space Z
2
⊗∧
2
Z
6
, and the set of isomor-
phism classes of pairs (R, M), where R is a nondegenerate cubic ring and M
is an equivalence class of balanced ideals of R having rank 2. Under this bijec-
tion, the discriminant of a pair of senary alternating 2-forms is equal to the
discriminant of the corresponding cubic ring.
Proof. Given a pair (R, M) as in the theorem, we first show how to con-
struct a corresponding pair of senary alternating 2-forms. Let again 1,ω,θ be
a normal basis for R, and let α
1
,α
2
, ,α
6
 denote an appropriately oriented
Z-basis for the rank 2 ideal M. By hypothesis, we may write
(17) det(α
i
,α
j
)=c
ij
+ b
ij
ω + a
ij
θ
HIGHER COMPOSITION LAWS II
877
for some 45 integers c
ij
, b
ij
, a
ij
satisfying
c
ij
= −c
ji
,b
ij
= −b
ji
,a
ij
= −a
ji
for all i, j ∈{1, 2, ,6}. Let A and B denote the 6 × 6 matrices (a
ij
) and
(b
ij
) respectively. Then (A, B) represents our desired pair of senary alternating
2-forms.
By construction, it is clear that changing the basis for M by an element
of SL
6
(Z) simply transforms (A, B) by that same element. Hence the SL
6
(Z)-
equivalence class of (A, B) is well-defined.
We wish to show that the mapping (R, M) → (A, B) is in fact a bijec-
tion. To this end, let us fix an element (A, B) ∈ Z
2
⊗∧
2
Z
6
, and consider the
system (17), which currently consists mostly of indeterminates. We show that
essentially all constants in this system are uniquely determined by (A, B).
First we claim the ring R is determined by (A, B). To show this, we
assume (3), and derive from (17) the identity
(18)
Pfaff(Ax −By)=N (M) · (ax
3
+ bx
2
y + cxy
2
+ dy
3
)
= ax
3
+ bx
2
y + cxy
2
+ dy
3
,
where we have used the hypothesis that N(M) = 1. It follows, as in the proof
of Theorem 2, that the ring R is determined by the pair (A, B).
To show that the constants c
ij
are determined, we use the identity
det(v
1
,v
3
) · det(v
2
,v
4
) = det(v
1
,v
2
) · det(v
3
,v
4
) + det(v
1
,v
4
) · det(v
2
,v
3
)
which holds for any four vectors v
1
,v
2
,v
3
,v
4
in the coordinate plane (a special
case of the Pl¨ucker relations). Since this identity holds over any ring, we have
in R the relations
(19) det(α
i
,α
k
)·det(α
j
,α

) = det(α
i
,α
j
)·det(α
k
,α

)+det(α
i
,α

)·det(α
j
,α
k
)
for i, j, k,  ∈{1, 2, ,6}. Expanding out these relations using (17), and equat-
ing the coefficients of 1, ω, and θ, leads to 45 linear and quadratic equations
in the c
ij
’s, in terms of the a
ij
’s and b
ij
’s. This system turns out to have a
unique solution, given by
(20) c
ij
= −

k,,m,n

ijkmn
123456

Pfaff(A
ijkl
) · Pfaff(B
ijmn
),
where we use

ijkmn
123456

to denote the sign of the permutation (i, j, k, , m, n)
of (1, 2, 3, 4, 5, 6). Thus the (integers) c
ij
in (17) are also uniquely determined
by (A, B ).
We claim that the Z-module M is now determined. Indeed, the values
of all determinants det(α
i
,α
j
) are determined by (17). Moreover, these deter-
minants satisfy the Pl¨ucker relations required of them as a result of (19). It
follows that the values of α
1
, ,α
6
are uniquely determined as elements of