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In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state. | Annals of Mathematics Higher composition laws III The parametrization of quartic rings By Manjul Bhargava Annals of Mathematics 159 2004 1329 1360 Higher composition laws III The parametrization of quartic rings By Manjul Bhargava 1. Introduction In the first two articles of this series we investigated various higher analogues of Gauss composition and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state. In the quadratic case one need only note that a quadratic ring i.e. any ring that is free of rank 2 as a Z-module is uniquely specified up to isomorphism by its discriminant and conversely given any discriminant D i.e. any integer congruent to 0 or 1 mod 4 there is a unique quadratic ring having discriminant D namely 1 I Z x x2 if D 0 S D Z 1 1 ỰD Z Z if D 1 is a square I Z D y D 2 otherwise. Thus we may say that quadratic rings are parametrized by the set D D E Z D 0 or 1 mod 4 . For a more detailed discussion of quadratic rings see 2 . The cubic case is slightly more complex in that cubic rings are not parametrized only by their discriminants indeed there may sometimes be several cubic orders having the same discriminant. The correct object parametrizing cubic rings i.e. rings free of rank 3 as Z-modules was first determined by Delone-Faddeev in their classic 1964 treatise on cubic irrationalities 8 . They showed that cubic rings are in bijective correspondence with GL2 Z -equivalence classes of integral binary cubic forms as follows. Given a binary cubic form f x y ax3 bx2y cxy2 dy3 with a b c d E Z one associates to f the ring R f having Z-basis 1 w1 w2 and multiplication table 2 A A 2 2 A ad ac bx 1 UX 2 bd dw1 cw2. 1330 MANJUL BHARGAVA One easily verifies that GL2 Z -equivalent binary cubic forms yield isomorphic rings .
