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Higher composition laws I: A new view on Gauss composition, and quadratic generalizations

# Higher composition laws I: A new view on Gauss composition, and quadratic generalizations,10.4007/annals.2004.159.217,Annals of Mathematics,Manjul Bha

Higher composition laws I: A new view on Gauss composition, and quadratic generalizations
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1. 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 SL2(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
Journal: Annals of Mathematics - ANN MATH , vol. 159, no. 1, pp. 217-250, 2004
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## Citation Context (8)

• ...perspective is given in the pioneering work of Bhargava [5], which has opened up new directions for broad generalizations of the classical theory...
• ...(2) ⇒ (3) can be deduced by direct computation using the characterization of composition given by Bhargava [5]...

### A. G. Earnest, et al. Multiplicative Properties of Integral Binary Quadratic Forms

• ...Recently Bhargava [1, 2, 3] developed the theory of integral orbits for those representations extensively and now significant contributions to such as algebraic number theory, computational number theory, or the theory of automorphic forms are being realized...

### Takashi Taniguchi. On parameterizations of rational orbits of some forms of prehomogeneou...

• ...As such, it is contained in the remarkable article [3] of M. Bhargava, who associates the quadratic forms of type f4 with classes of cubic forms...

### Francesca Aicardi, et al. On binary quadratic forms with the semigroup property

• ...In this section, we follow Bhargava [4], and study the scheme C over Z which satisfies C(R) = R 2 ⊗ R 2 ⊗ R 2 for any ring R. If c ∈ C(R),...
• ...Following the appendix in [4], it is useful to know that projective cubes can be put into a particularly nice form via the action of = SL 2 (Z) 3 :...
• ...A beautiful generalization of the above result is a consequence of Theorem 11 in Bhargava [4]:...
• ...We can deduce from results of Bhargava in [4], that every 2 by 2 by 2 cube also yields a QT-structure as well...
• ...Suppose that the cube c is in normal form as before, with discriminant D 6 0, and let R(D) denote the quadratic ring of discriminant D. Associated to the cube c, we get three invertible oriented ideal classes I1, I2, I3. Let � denote the free Z-module of rank 2, with basis λ, � . By results in the Appendix of [4], there exist Z-module isomorphisms from � to I1, I2, I3 (choosing appropriate representatives for these ideal classes), such ...

### Martin H. Weissman. D 4 modular forms

• ...After Wright-Yukie, the arithmetic theory of Z-orbits was first handled by Kable [K00] for one special case, and recently being developed greatly by Bhargava [B04a, B04b, B04c, B05]...

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