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Annals of Mathematics, 159 (2004), 217–
Higher composition laws I:
A new view on Gauss composition,
and quadratic generalizations
By Manjul Bhargava
- Introduction
Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow. Even today, two centuries later, this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders. It is hence only natural to ask whether higher analogues of this composi- tion law exist that could shed light on the structure of other algebraic number rings and fields. This article forms the first of a series of four articles in which our aim is precisely to develop such “higher composition laws”. In fact, we show that Gauss’s law of composition is only one of at least fourteen compo- sition laws of its kind which yield information on number rings and their class groups. In this paper, we begin by deriving a general law of composition on 2× 2 × 2 cubes of integers, from which we are able to obtain Gauss’s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves. We also obtain from this composition law on 2 × 2 × 2 cubes four further new laws of composition. These laws of composition are defined on 1) binary cubic forms, 2) pairs of binary quadratic forms, 3) pairs of quaternary alternating 2-forms, and 4) senary (six-variable) alternating 3-forms. More precisely, Gauss’s theorem states that the set of SL 2 (Z)-equivalence classes of primitive binary quadratic forms of a given discriminant D has an inherent group structure. The five other spaces of forms mentioned above (including the space of 2 × 2 × 2 cubes) also possess natural actions by special linear groups over Z and certain products thereof. We prove that, just like Gauss’s space of binary quadratic forms, each of these group actions has the following remarkable properties. First, each of these six spaces possesses only a single polynomial invariant for the corresponding group action, which we call the discriminant. This discriminant invariant is found to take only values that
218 MANJUL BHARGAVA
are 0 or 1 (mod 4). Second, there is a natural notion of projectivity for elements in these spaces, which reduces to the notion of primitivity in the case of binary quadratic forms. Finally, for each of these spaces L, the set Cl(L; D) of orbits of projective elements having a fixed discriminant D is naturally equipped with the structure of a finite abelian group. The six composition laws mentioned above all turn out to have natural interpretations in terms of ideal classes of quadratic rings. We prove that the law of composition on 2 × 2 × 2 cubes of discriminant D gives rise to groups isomorphic to Cl+(S) × Cl+(S), where Cl+(S) denotes the narrow class group of the quadratic order S of discriminant D. This interpretation of the space of 2 × 2 × 2 cubes then specializes to give the narrow class group in Gauss’s case and in the cases of pairs of binary quadratic forms and pairs of quaternary alternating 2-forms, and yields roughly the 3-part of the narrow class group in the case of binary cubic forms. Finally, it gives the trivial group in the case of six-variable alternating 3-forms, yielding the interesting consequence that, for any fundamental discriminant D, there is exactly one integral senary 3-form E ∈ ∧^3 Z^6 having discriminant D (up to SL 6 (Z)-equivalence). We note that many of the spaces we derive in this series of articles were previously considered over algebraically closed fields by Sato-Kimura [7] in their monumental work classifying prehomogeneous vector spaces. Over other fields such as the rational numbers, these spaces were again considered in the important work of Wright-Yukie [9], who showed that generic rational orbits in these spaces correspond to ´etale extensions of degrees 1, 2, 3, 4, or 5. Our approach differs from previous work in that we consider orbits over the integers Z; as we shall see, the integer orbits have an extremely rich structure, extending Gauss’s work on the space of binary quadratic forms to various other spaces of forms. The organization of this paper is as follows. Section 2 forms an ex- tended introduction in which we describe, in an elementary manner, the above- mentioned six composition laws and the elegant properties which uniquely de- termine them. In Section 3 we describe how to rephrase these six composition laws in the language of ideal classes of quadratic orders, when the discriminant is nonzero; we use this new formulation to provide proofs of the assertions of Section 2 as well as to gain an understanding of the nonprojective elements of these spaces in terms of nonprojective ideal classes. In Section 4, we conclude by discussing the mysterious relationship between our composition laws and the exceptional Lie groups.
Remarks on terminology and notation. An n-ary k-ic form is a homoge- neous polynomial in n variables of degree k. For example, a binary quadratic form is a function of the form f (x, y) = ax^2 + bxy + cy^2 for some coefficients a, b, c. We will denote by (SymkZn)∗^ the
(n+k− 1 k
-dimensional lattice of n-ary
220 MANJUL BHARGAVA
into the 2 × 2 matrices
M 1 =
[
a b c d
]
, N 1 =
[
e f g h
]
or into
M 2 =
[
a c e g
]
, N 2 =
[
b d f h
]
or
M 3 =
[
a e b f
]
, N 3 =
[
c g d h
]
Our action of Γ is defined so that, for any 1 ≤ i ≤ 3, an element ( r st u ) in the ith^ factor of SL 2 (Z) acts on the cube A by replacing (Mi, Ni) by (rMi + sNi, tMi + uNi). The actions of these three factors of SL 2 (Z) in Γ commute with each other; this is analogous to the fact that row and column operations on a rectangular matrix commute. Hence we obtain a natural action of Γ on C 2. Now given any cube A ∈ C 2 as above, let us construct a binary quadratic form Qi = QAi for 1 ≤ i ≤ 3, by defining
Qi(x, y) = −Det(Mix − Niy).
Then note that the form Q 1 is invariant under the action of the subgroup {id} × SL 2 (Z) × SL 2 (Z) ⊂ Γ, because this subgroup acts only by row and column operations on M 1 and N 1 and hence does not change the value of −Det(M 1 x − N 1 y). The remaining factor of SL 2 (Z) acts in the standard way on Q 1 , and it is well-known that this action has exactly one polynomial invari- ant^1 , namely the discriminant Disc(Q 1 ) of Q 1 (see, e.g., [6]). Thus the unique polynomial invariant for the action of Γ = SL 2 (Z) × SL 2 (Z) × SL 2 (Z) on its representation Z^2 ⊗ Z^2 ⊗ Z^2 is given simply by Disc(Q 1 ). Of course, by the same reasoning, Disc(Q 2 ) and Disc(Q 3 ) must also be equal to this same invari- ant up to scalar factors. A symmetry consideration (or a quick calculation!) shows that in fact Disc(Q 1 ) = Disc(Q 2 ) = Disc(Q 3 ); we denote this common value simply by Disc(A). Explicitly, we find
Disc(A) = a^2 h^2 + b^2 g^2 + c^2 f 2 + d^2 e^2 −2(abgh + cdef + acf h + bdeg + aedh + bf cg) + 4(adf g + bceh).
(^1) We use throughout the standard abuse of terminology “has one polynomial invariant” to mean that the corresponding polynomial invariant ring is generated by one element.
HIGHER COMPOSITION LAWS I 221
2.2. Gauss composition revisited. We have seen that every cube A in C 2 gives three integral binary quadratic forms QA 1 , QA 2 , QA 3 all having the same discriminant. Inspired by the group law on elliptic curves, let us define an addition axiom on the set of (primitive) binary quadratic forms of a fixed discriminant D by declaring that, for all triplets of primitive quadratic forms QA 1 , QA 2 , QA 3 arising from a cube A of discriminant D,
The Cube Law. The sum of QA 1 , QA 2 , QA 3 is zero. More formally, we consider the free abelian group on the set of primitive binary quadratic forms of discriminant D modulo the subgroup generated by all sums [QA 1 ] + [QA 2 ] + [QA 3 ] with QAi as above. One basic and beautiful consequence of this axiom of addition is that forms that are SL 2 (Z)-equivalent automatically become “identified”, for the following reason. Suppose that γ = γ 1 × id × id ∈ Γ, and that A gives rise to the three quadratic forms Q 1 , Q 2 , Q 3. Then A′^ = γA gives rise to the three quadratic forms Q′ 1 , Q 2 , Q 3 , where Q′ 1 = γ 1 Q 1. Now the Cube Law implies that the sum of Q 1 , Q 2 , Q 3 is zero, and also that the sum of Q′ 1 , Q 2 , Q 3 is zero. Therefore Q 1 and Q′ 1 become identified, and thus we may view the Cube Law as descending to a law of addition on SL 2 (Z)-equivalence classes of forms of a given discriminant. In fact, with an appropriate choice of identity, this simple relation imposed by the Cube Law turns the space of SL 2 (Z)-equivalence classes of primitive binary quadratic forms of discriminant D into a group! More precisely, for a binary quadratic form Q let us use [Q] to denote the SL 2 (Z)-equivalence class of Q. Then we have the following theorem.
Theorem 1. Let D be any integer congruent to 0 or 1 (mod 4), and let Qid,D be any primitive binary quadratic form of discriminant D such that there is a cube A 0 with QA 1 0 = QA 2 0 = QA 3 0 = Qid,D. Then there exists a unique group law on the set of SL 2 (Z)-equivalence classes of primitive binary quadratic forms of discriminant D such that:
(a) [Qid,D] is the additive identity; (b) For any cube A of discriminant D such that QA 1 , QA 2 , QA 3 are primitive, we have
[QA 1 ] + [QA 2 ] + [QA 3 ] = [Qid,D].
Conversely, given Q 1 , Q 2 , Q 3 with [Q 1 ] + [Q 2 ] + [Q 3 ] = [Qid,D], there exists a cube A of discriminant D, unique up to Γ-equivalence, such that QA 1 = Q 1 , QA 2 = Q 2 , and QA 3 = Q 3.
The most natural choice of identity element in Theorem 1 is
Qid,D = x^2 −
D
y^2 or Qid,D = x^2 − xy +
1 − D
(2) y^2
HIGHER COMPOSITION LAWS I 223
Theorem 2. Let D be any integer congruent to 0 or 1 (mod 4), and let Aid,D be the triply-symmetric cube defined by (3). Then there exists a unique group law on the set of Γ-equivalence classes of projective cubes A of discrim- inant D such that:
(a) [Aid,D] is the additive identity; (b) For i = 1, 2 , 3, the maps [A] → [QAi ] yield group homomorphisms to Cl
(Sym^2 Z^2 )∗; D
We note again that other identity elements could have been chosen in Theorem 2. However, for concreteness, we choose Aid,D as in (3) once and for all, since this choice determines the choice of identity element in all other compositions (including Gauss composition). Theorem 2 is easily deduced from Theorem 1. In fact, addition of cubes may be defined in the following manner. Let A and A′^ be any two projec- tive cubes having discriminant D; since ([QA 1 ] + [QA ′ 1 ]) + ([Q A 2 ] + [Q A′ 2 ])+ ([QA 3 ] + [QA ′ 3 ]) = [Qid,D] in Cl
(Sym^2 Z^2 )∗; D
, the existence of a cube A′′^ with
[QA ′′ i ] = [Q A i ] + [Q A′ i ] for 1^ ≤^ i^ ≤^ 3 and its uniqueness up to Γ-equivalence follows from the last part of Theorem 1. We define the composition of [A] and [A′] by setting [A] + [A′] = [A′′]. We denote the set of Γ-equivalence classes of projective cubes of discrim- inant D, equipped with the above group structure, by Cl(Z^2 ⊗ Z^2 ⊗ Z^2 ; D).
2.4. Composition of binary cubic forms. The above law of composition on cubes also leads naturally to a law of composition on (SL 2 (Z)-equivalence classes of) integral binary cubic forms px^3 + 3qx^2 y + 3rxy^2 + sy^3. For just as one frequently associates to a binary quadratic form px^2 + 2qxy + ry^2 the symmetric 2 × 2 matrix [ p q q r
]
one may naturally associate to a binary cubic form px^3 + 3qx^2 y + 3rxy^2 + sy^3 the triply-symmetric 2 × 2 × 2 matrix
p q
q (^) r
q (^) r
r s
224 MANJUL BHARGAVA
Using Sym^3 Z^2 to denote the space of binary cubic forms with triplicate central coefficients, the above association of px^3 + 3qx^2 y + 3rxy^2 + sy^3 with the cube (4) corresponds to the natural inclusion
ι : Sym^3 Z^2 → Z^2 ⊗ Z^2 ⊗ Z^2
of the space of triply-symmetric cubes into the space of cubes. We call a binary cubic form C(x, y) = px^3 + 3qx^2 y + 3rxy^2 + sy^3 projective if the corresponding triply-symmetric cube ι(C) given by (4) is projective. In
this case, the three forms Qι 1 ( C), Qι 2 ( C), Qι 3 ( C)are all equal to the Hessian
H(x, y) = (q^2 − pr)x^2 + (ps − qr)xy + (r^2 − qs)y^2 = −
∣∣ Cxx Cxy Cyx Cyy
hence C is projective if and only if H is primitive, i.e., if gcd(q^2 − pr, ps − qr, r^2 − qs) = 1. The preimages of the identity cubes (3) under ι are given by
Cid,D = 3x^2 y +
D
y^3 or Cid,D = 3x^2 y + 3xy^2 +
D + 3
(6) y^3
in accordance with whether D ≡ 0 (mod 4) or D ≡ 1 (mod 4). Denoting the SL 2 (Z)-equivalence class of C ∈ Sym^3 Z^2 by [C], we have the following theorem.
Theorem 3. Let D be any integer congruent to 0 or 1 modulo 4, and let Cid,D be given as in (6). Then there exists a unique group law on the set of SL 2 (Z)-equivalence classes of projective binary cubic forms C of discriminant D such that:
(a) [Cid,D] is the additive identity; (b) The map given by [C] → [ ι(C) ] is a group homomorphism to Cl(Z^2 ⊗ Z^2 ⊗ Z^2 ; D).
We denote the set of equivalence classes of projective binary cubic forms of discriminant D, equipped with the above group structure, by Cl(Sym^3 Z^2 ; D).
2.5. Composition of pairs of binary quadratic forms. The group law on binary cubic forms of discriminant D was obtained by imposing a symmetry condition on the group of 2 × 2 × 2 cubes of discriminant D, and determining that this symmetry was preserved under the group law. Rather than imposing a threefold symmetry, one may instead impose only a twofold symmetry. This leads to cubes taking the form
226 MANJUL BHARGAVA
taking twofold symmetric projective cubes B ∈ Z^2 ⊗ Sym^2 Z^2 to their third associated quadratic form QB 3 , yields an isomorphism of groups.^3
2.6. Composition of pairs of quaternary alternating 2-forms. Instead of imposing conditions of symmetry, one may impose conditions of skew- symmetry on cubes using a certain “fusion” process. To define these skew- symmetrizations, let us view our original space Z^2 ⊗ Z^2 ⊗ Z^2 as the space of Z-trilinear maps L 1 × L 2 × L 3 → Z, where L 1 , L 2 , L 3 are Z-modules of rank 2 (namely, the Z-duals of the three factors Z^2 in Z^2 ⊗ Z^2 ⊗ Z^2 ). Then given such a trilinear map
φ : L 1 × L 2 × L 3 → Z
in Z^2 ⊗ Z^2 ⊗ Z^2 , one may naturally construct another Z-trilinear map
φ¯ : L 1 × (L 2 ⊕ L 3 ) × (L 2 ⊕ L 3 ) → Z
that is skew-symmetric in the second and third variables; this map φ¯ = id ⊗ ∧ 2 , 2 (φ) is given by
φ¯ (r, (s, t), (u, v)) = φ(r, s, v) − φ(r, u, t).
Thus we have a natural Z-linear mapping
(9) id ⊗ ∧ 2 , 2 : Z^2 ⊗ Z^2 ⊗ Z^2 → Z^2 ⊗ ∧^2 (Z^2 ⊕ Z^2 ) = Z^2 ⊗ ∧^2 Z^4
taking 2× 2 ×2 cubes to pairs of alternating 2-forms in four variables. Explicitly, in terms of fixed bases for L 1 , L 2 , L 3 , this mapping is given by
a (^) b
c (^) d
e f
g (^) h
→
a b c d −a −c −b −d
e f g h −e −g −f −h
Let Γ = SL 2 (Z) × SL 2 (Z) × SL 2 (Z) as before, and set Γ′^ = SL 2 (Z) × SL 4 (Z). Then it is clear from our description that two elements in the same Γ-equivalence class in Z^2 ⊗ Z^2 ⊗ Z^2 will map by (9) (or (10)) to the same Γ′-equivalence class in Z^2 ⊗ ∧^2 Z^4. More remarkably, as we will prove in Sec- tion 3.6, the map (9) is surjective on the level of equivalence classes; that is,
(^3) That these two spaces (Sym (^2) Z (^2) )∗ (^) and Z (^2) ⊗ Sym (^2) Z (^2) carry similar information is a re- flection of the fact that, in the language of prehomogeneous vector spaces, Sym^2 Z^2 is a reduced form of the space Z^2 ⊗ Sym^2 Z^2 , i.e., is the smallest space that can be obtained from Z^2 ⊗ Sym^2 Z^2 by what are called “castling transforms” (cf. [7]).
HIGHER COMPOSITION LAWS I 227
any element v ∈ Z^2 ⊗ ∧^2 Z^4 can be transformed by an element of Γ′^ to lie in the image of (9) or (10). We say that an element F ∈ Z^2 ⊗ ∧^2 Z^4 is projective if it is Γ′-equivalent to (id ⊗ ∧ 2 , 2 )(A) for some projective cube A. Now to any pair F = (M, N ) ∈ Z^2 ⊗ ∧^2 Z^4 of alternating 4 × 4 matrices, one can naturally associate a binary quadratic form Q = QF^ given by
−Q(x, y) = Pfaff(M x − N y) =
Det(M x − N y),
where, as is customary, we choose the sign of the Pfaffian so that
Pfaff
([
I
−I
])
We obtain therefore an SL 2 -equivariant map
(11) Z^2 ⊗ ∧^2 Z^4 → (Sym^2 Z^2 )∗.
One easily checks that the coefficients of the covariant Q(x, y) give a complete set of polynomial invariants for the action of SL 4 (Z) on Z^2 ⊗ ∧^2 Z^4. Hence the space of elements (M, N ) ∈ Z^2 ⊗ ∧^2 Z^4 possesses a unique polynomial invariant for the action of Γ′^ = SL 2 (Z) × SL 4 (Z), namely
Disc(Pfaff(M x − N y)).
We call this unique, degree 4 invariant the discriminant Disc(F ) of F. It is evident from the explicit formula (10) that the linear map (9) is discriminant- preserving. Since the mapping (9) is surjective on the level of equivalence classes, and the Γ-equivalence classes of projective cubes having discriminant D form a group, we might suspect that the Γ′-equivalence classes of projective elements in Z^2 ⊗ ∧^2 Z^4 having discriminant D also possess a natural composition law. In fact, this is the case; denoting by [F ] the Γ′-equivalence class of F , we have the following theorem.
Theorem 5. Let D be any integer congruent to 0 or 1 modulo 4, and let Fid,D = id ⊗ ∧ 2 , 2 (Aid,D). Then there exists a unique group law on the set of Γ′-equivalence classes of projective pairs of quaternary alternating 2-forms F of discriminant D such that:
(a) [Fid,D] is the additive identity; (b) The map given by [A] → [id ⊗ ∧ 2 , 2 (A)] is a group homomorphism from Cl(Z^2 ⊗ Z^2 ⊗ Z^2 ; D);
(b′) The map given by [F ] → [QF^ ] is a group homomorphism to Cl
(Sym^2 Z^2 )∗; D
In fact, either (b) or (b′) would be sufficient in Theorem 5 to specify the desired group structure. We denote the set of Γ′-equivalence classes of
HIGHER COMPOSITION LAWS I 229
We say that an element E ∈ ∧^3 Z^6 is projective if it is SL 6 (Z)-equivalent to ∧ 2 , 2 , 2 (A) for some projective cube A. Because the projective classes of cubes in Z^2 ⊗ Z^2 ⊗ Z^2 of discriminant D possess a group law, and the map (12) is surjective on equivalence classes, we may reasonably expect that (as in the case of Z^2 ⊗ ∧^2 Z^4 ) the projective classes in ∧^3 Z^6 of discriminant D should also turn into a group, defined by a pair of conditions (a) and (b) analogous to those presented in Theorems 1–5. This is indeed the case. However, as we will prove in Section 3.7 from the point of view of mod- ules over quadratic orders, this resulting group Cl(∧^3 Z^6 ; D) always consists of exactly one element! Thus it becomes rather unnecessary to state a theorem for ∧^3 Z^6 akin to Theorems 1–5. Instead, we have the following theorem.
Theorem 7. Let D be any integer congruent to 0 or 1 modulo 4. Then the set Cl(∧^3 Z^6 ; D) consists only of the single element [Eid,D] = [∧ 2 , 2 , 2 (Aid,D)]. If furthermore D is a fundamental discriminant,^5 then all six - variable alternating 3-forms with discriminant D are projective, and hence up to SL 6 (Z)-equivalence there is exactly one senary alternating 3-form of discriminant D.
To summarize Section 2, we have natural, discriminant-preserving arrows
Sym^3 Z^2 Z^2 ⊗ Sym^2 Z^2
Z 2 ⊗ Z 2 ⊗ Z 2
(Sym^2 Z^2 )∗^ Z^2 ⊗ ∧^2 Z^4
∧^3 Z^6
leading to the group homomorphisms
Cl(Sym^3 Z^2 ; D) Cl(Z^2 ⊗ Sym^2 Z^2 ; D)
Cl(Z^2 ⊗ Z^2 ⊗ Z^2 ; D)
Cl
(Sym^2 Z^2 )∗; D
Cl(Z (^2) ⊗ ∧ (^2) Z (^4) ; D)
Cl(∧^3 Z^6 ; D)
where the central two arrows to Cl
(Sym^2 Z^2 )∗; D
are in fact isomorphisms, and the bottom group Cl(∧^3 Z^6 ; D) is trivial.
(^5) Recall that an integer D is called a fundamental discriminant if it is square-free and 1 (mod 4) or it is four times a square-free integer that is 2 or 3 (mod 4). Asymptotically, 6 /π^2 ≈ 61% of all discriminants are fundamental.
230 MANJUL BHARGAVA
- Relations with ideal classes in quadratic orders
The integral orbits of the six spaces discussed in the previous section each have natural interpretations in terms of quadratic orders.
3.1. The parametrization of quadratic rings. In the first four papers of this series, we will be interested in studying commutative rings R with unit whose underlying additive group is Zn^ for n = 2, 3 , 4 , and 5; such rings are called quadratic, cubic, quartic, and quintic rings respectively.^6 The proto- typical example of such a ring is, of course, an order in a number field of degree at most 5. To any such ring of rank n we may attach the trace function Tr : R → Z, which assigns to an element α ∈ R the trace of the endomorphism
R −×→ Rα. The discriminant Disc(R) of such a ring R is then defined as the determinant det(Tr(αiαj )) ∈ Z, where {αi} is any Z-basis of R. It is a classical fact, due to Stickelberger, that a ring having finite rank as a Z-module must have discriminant congruent to 0 or 1 (mod 4). In the case of rank 2, this is easy to see: such a ring must have Z-basis of the form 〈 1 , τ 〉, where τ satisfies a quadratic τ 2 + rτ + s = 0 with r, s ∈ Z. The discriminant of this ring is then computed to be r^2 − 4 s, which is congruent to 0 or 1 modulo 4. Conversely, given any integer D ≡ 0 or 1 (mod 4) there is a unique quadratic ring S(D) having discriminant D (up to isomorphism), given by
S(D) =
Z[x]/(x^2 ) if D = 0, Z · (1, 1) +
D(Z ⊕ Z) if D ≥ 1 is a square, Z[(D +
D)/2] otherwise;
explicitly, S(D) has Z-basis 〈 1 , τ 〉 where multiplication is determined by the law
τ 2 =
D
or τ 2 =
D − 1
(14) + τ
in accordance with whether D ≡ 0 (mod 4) or D ≡ 1 (mod 4).^7 Therefore, if we denote by D the set of elements of Z that are congruent to 0 or 1 (mod 4), we may say that isomorphism classes of quadratic rings are parametrized by D. There is a slight problem with this latter parametrization, however, in that all quadratic rings have two automorphisms, whereas, at least as stated, corresponding elements of D do not. As a result, the above construction
(^6) In subsequent articles, we will turn our attention to noncommutative rings. (^7) This case distinction, which will persist throughout the paper, could be avoided by
writing S(D) as Z + Zτ where τ is the root of τ 2 − Dτ + D^24 − D= 0, or of any quadratic τ 2 + rτ + s = 0 with r^2 − 4 s = D; but then one would also have the variables r, s, or D in all the formulas, so we have preferred instead to fix the choice r ∈ { 0 , 1 }.
232 MANJUL BHARGAVA
of Kn^ by a matrix of positive determinant. Any other basis is said to be negatively oriented. Finally, we say that a quadratic ring is nondegenerate if its discriminant is nonzero, i.e., if it is not isomorphic to the (degenerate) quadratic ring S(0). Similarly, we say that an element v ∈ L—where L is any one of the six spaces (Sym^2 Z^2 )∗, Z^2 ⊗Z^2 ⊗Z^2 , Sym^3 Z^2 , Z^2 ⊗Sym^2 Z^2 , Z^2 ⊗∧^2 Z^4 , or ∧^3 Z^6 introduced in Section 2—is nondegenerate if its discriminant Disc(v) is nonzero. In the forthcoming sections, we show that the orbits of nondegenerate elements in these six spaces may be completely classified in terms of certain special types of ideal classes in nondegenerate quadratic rings. We begin by recalling briefly the classical case of binary quadratic forms.
3.2. The case of binary quadratic forms. As is well-known, the group Cl
(Sym^2 Z^2 )∗; D
is almost, but not quite the same as, the ideal class group of the unique quadratic order S of discriminant D. To make up for the slight discrepancy, it is necessary to introduce the notion of narrow class group, which may be defined as the group Cl+(S) of oriented ideal classes. More precisely, an oriented ideal is a pair (I, ε), where I is any (fractional) ideal of S in K = S ⊗ Q having rank 2 as a Z-module, and ε = ±1 gives the orientation of I. Multiplication of oriented ideals is defined componentwise, and the norm of an oriented ideal (I, ε) is defined to be ε·|L/I|·|L/S|−^1 , where L is any lattice in K containing both S and I. For an element κ ∈ K, the product κ · (I, ε) is defined to be the ideal (κ I, sgn(N (κ))ε). Two oriented ideals (I 1 , ε 1 ) and (I 2 , ε 2 ) are said to be in the same oriented ideal class if (I 1 , ε 1 ) = κ · (I 2 , ε 2 ) for some invertible κ ∈ K. With these notions, the narrow class group can then be defined as the group of invertible oriented ideals modulo multiplication by invertible scalars κ ∈ K (equivalently, modulo the subgroup consisting of invertible principal oriented ideals ((κ), sgn(N (κ)))). The elements of this group are thus the invertible oriented ideal classes. In practice, we shall denote an oriented ideal (I, ε) simply by I, with the orientation ε = ε(I) on I being understood.^9 We may now state the precise relation between equivalence classes of bi- nary quadratic forms and ideal classes of quadratic orders.
Theorem 9. There is a canonical bijection between the set of nondegen- erate SL 2 (Z)-orbits on the space (Sym^2 Z^2 )∗^ of integer - valued binary quadratic forms, and the set of isomorphism classes of pairs (S, I), where S is a nonde- generate oriented quadratic ring and I is a (not necessarily invertible) oriented
(^9) Traditionally, the narrow class group is considered only for quadratic orders S of positive discriminant, and is defined as the group of invertible ideals of S modulo the subgroup of invertible principal ideals that are generated by elements of positive norm. We prefer our definition here since it gives the correct notion also when D < 0.
HIGHER COMPOSITION LAWS I 233
ideal class of S. Under this bijection, the discriminant of a binary quadratic form equals the discriminant of the corresponding quadratic ring.
Restricting the above result to the set of primitive quadratic forms, and noting that, in the above bijection, primitive binary quadratic forms corre- spond to invertible ideal classes, we obtain the following group isomorphism.
Theorem 10. The bijection of Theorem 9 restricts to a correspondence Cl
(Sym^2 Z^2 )∗; D
↔ Cl+(S(D)),
which is an isomorphism of groups.
We remark—although it will not be used in this paper—that the usual (as opposed to narrow) ideal class group may be obtained as the set of GL 2 (Z)- (rather than SL 2 (Z)-) equivalence classes of primitive binary quadratic forms, except that we must then let an element α ∈ GL 2 (Z) act on a form Q by Q → (^) det(^1 α) · αQ.
Theorem 9 is known in the indefinite case, while the general definite case follows easily from the known case of positive definite quadratic forms. We will give proofs of Theorems 9 and 10 in a more general context in the next section.
3.3. The case of 2 × 2 × 2 cubes. We now turn to the general case of 2 × 2 × 2 cubes. Before stating the result, we make some definitions. Let S be the quadratic ring of discriminant D, and let K = S ⊗ Q be the corresponding quadratic algebra over Q. We say that a triple (I 1 , I 2 , I 3 ) of oriented ideals of S is balanced if I 1 I 2 I 3 ⊆ S and N (I 1 )N (I 2 )N (I 3 ) = 1. Also, we define two balanced triples (I 1 , I 2 , I 3 ) and (I 1 ′, I 2 ′, I 3 ′) of ideals of S to be equivalent if I 1 = κ 1 I 1 ′, I 2 = κ 2 I 2 ′, I 3 = κ 3 I 3 ′ for some elements κ 1 , κ 2 , κ 3 ∈ K. (In particular, we must have N (κ 1 κ 2 κ 3 ) = 1.) For example, if S is Dedekind, then an equivalence class of balanced triples means simply a triple of narrow ideal classes whose product is the principal class. Our main result on 2 × 2 × 2 cubes is then as follows:
Theorem 11. There is a canonical bijection between the set of nondegen- erate Γ-orbits on the space Z^2 ⊗Z^2 ⊗Z^2 of 2 × 2 × 2 integer cubes, and the set of isomorphism classes of pairs (S, (I 1 , I 2 , I 3 )), where S is a nondegenerate ori- ented quadratic ring and (I 1 , I 2 , I 3 ) is an equivalence class of balanced triples of oriented ideals of S. Under this bijection, the discriminant of an integer cube equals the discriminant of the corresponding quadratic ring.
Proof. For a balanced triple (I 1 , I 2 , I 3 ) of ideals of an oriented quadratic order S = S(D) as in the theorem, we first show how to construct a correspond- ing 2 × 2 × 2 cube. In accordance with whether D = Disc(S) is congruent to 0
HIGHER COMPOSITION LAWS I 235
Aid,D by T × {e} × {e} ∈ Γ. The quadratic form QA 2 (or QA 3 ) is thus seen to multiply by a factor of det(T ) = N (I 1 ), so that the discriminant of A becomes multiplied by a factor of N (I 1 )^2. In a similar manner, if I 2 and I 3 are also changed to general S-ideals, this will introduce factors of N (I 2 )^2 and N (I 3 )^2 in (16), thus proving the identity for general I 1 , I 2 , I 3. Now by assumption we have N (I 1 )N (I 2 )N (I 3 ) = 1, so that
(17) Disc(A) = Disc(S),
and hence S is indeed determined by A to be S(Disc(A)). Next, by the associativity and commutativity of S, we must have
αiβj γk · αi′^ βj′^ γk′^ = αi′^ βj γk · αiβj′^ γk′^ = αiβj′^ γk · αi′^ βj γk′^ = αiβj γk′^ · αi′^ βj′^ γk
for all 1 ≤ i, i′, j, j′, k, k′^ ≤ 2. Expanding out these identities using (15), and then equating all coefficients of 1 and τ , yield 18 (linear and quadratic) equa- tions in the eight variables cijk in terms of the aijk. We find that this system, together with the condition N (I 1 )N (I 2 )N (I 3 ) > 0, has a unique solution, given by
cijk = (i′^ −i)(j′^ −j)(k′^ −k) ·
[
ai′jkaij′kaijk′^ + 12 aijk(aijkai′j′k′^ −ai′jkaij′k′^ −aij′kai′jk′^ −aijk′^ ai′j′k)
]
− 12 aijk ε
with {i, i′} = {j, j′} = {k, k′} = { 1 , 2 }, and where ε = 0 or 1 in accordance with whether D ≡ 0 (mod 4) or D ≡ 1 (mod 4). A quick congruence check shows that the solutions for the cijk are necessarily integral! Therefore, the cijk’s in (15) are also uniquely determined by the cube A. We must still determine the existence of αi, βj , γk ∈ S yielding the desired aijk and cijk’s in (15). It is clear that the pair (α 1 , α 2 ) (similarly (β 1 , β 2 ), (γ 1 , γ 2 )) is uniquely determined—up to a nonzero scaling factor in K—by the equations (15). For example, given any fixed 1 ≤ j, k ≤ 2 for which c 1 jk +a 1 jkτ and c 2 jk + a 2 jkτ are invertible in K, we have
(19) α 1 βj γk(c 2 jk + a 2 jkτ ) = α 2 βj γk(c 1 jk + a 1 jkτ ),
so the ratio α 1 : α 2 is determined, and we may let, e.g., α 1 = c 1 jk + a 1 jkτ and α 2 = c 2 jk + a 2 jkτ. That this ratio α 1 : α 2 as determined by (19) is independent of j, k (up to a constant factor) follows from the associative laws (18) that have been forced upon the system (15). The pair (β 1 , β 2 ) can be similarly determined up to scalars in K, and then (γ 1 , γ 2 ) is completely determined by (α 1 , α 2 ) and (β 1 , β 2 ). Hence the triple (I 1 , I 2 , I 3 ) is completely determined up to equivalence. Thus we must show only that the Z-modules I 1 = 〈α 1 , α 2 〉, I 2 = 〈β 1 , β 2 〉, I 3 = 〈γ 1 , γ 2 〉 as determined above actually form ideals of S. In fact, it is
236 MANJUL BHARGAVA
possible to determine the precise S-module structures of I 1 , I 2 , I 3. Let Q 1 , Q 2 , Q 3 be the three quadratic forms associated to A as in Section 2.1, where we write Qi = pix^2 + qixy + riy^2. Then a short calculation using explicit expressions for αi, βj , γk as above shows that
τ · α 1 = q^12 + ε· α 1 + p 1 · α 2 , −τ · α 2 = r 1 · α 1 + q^1 − 2 ε· α 2
where again ε = 0 or 1 in accordance with whether D ≡ 0 or 1 (mod 4), and where the module structures of I 2 = 〈β 1 , β 2 〉 and I 3 = 〈γ 1 , γ 2 〉 are given analogously in terms of the forms Q 2 and Q 3 respectively. In particular, we conclude that I 1 , I 2 , I 3 are indeed ideals of S. We have now determined all the indeterminates in (15), having started only with the value of the cube A. It follows that there is exactly one pair (S, (I 1 , I 2 , I 3 )) up to equivalence that yields the cube A under the mapping (S, (I 1 , I 2 , I 3 )) → A; this completes the proof.
Note that the above discussion makes the bijection of Theorem 11 very precise. Given a quadratic ring S and a balanced triple (I 1 , I 2 , I 3 ) of ideals in S, the corresponding cube A = (aijk) is obtained from equations (15). Conversely, given a cube A ∈ Z^2 ⊗ Z^2 ⊗ Z^2 , the ring S is determined by (17); bases for the ideal classes I 1 , I 2 , I 3 in S are obtained from (15), and the S-module structures of I 1 , I 2 , and I 3 are given by (20). Let us define a balanced triple (I 1 , I 2 , I 3 ) of ideals of S to be projective if I 1 , I 2 , I 3 are projective as S-modules. Then there is a natural group law on the set of equivalence classes of projective balanced triples of ideals of a ring S. Namely, for any two such balanced triples (I 1 , I 2 , I 3 ) and (I 1 ′, I′ 2 , I 3 ′), define their composition to be the (balanced) triple (I 1 I 1 ′, I 2 I′ 2 , I 3 I 3 ′). This group of equivalence classes of projective balanced triples is naturally isomorphic to Cl+(S) × Cl+(S), via the map (I 1 , I 2 , I 3 ) → (I 1 , I 2 ). Restricting Theorem 11 to the set of projective elements of C 2 , and noting that projective cubes give rise to balanced triples of projective ideals, yields the following group isomorphism.
Theorem 12. The bijection of Theorem 11 restricts to a correspondence Cl(Z^2 ⊗ Z^2 ⊗ Z^2 ; D) ↔ Cl+(S(D)) × Cl+(S(D))
which is an isomorphism of groups.
That primitive binary quadratic forms and projective ideal classes are in one-to-one correspondence (the case of Gauss) is of course recovered as a special case. Indeed, a short calculation shows that the norm forms of I 1 , I 2 , I 3 as given by Theorem 11 are simply QA 1 , QA 2 , QA 3 , which are the three quadratic forms associated to A. Thus we have also proved Theorems 1, 2, 9, and 10.
