TAILIEUCHUNG - Đề tài " Higher composition laws I: A new view on Gauss composition, and quadratic generalizations "

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 composition law exist that could shed light on the structure of other algebraic number rings and fields. . | Annals of Mathematics Higher composition laws I A new view on Gauss composition and quadratic generalizations By Manjul Bhargava Annals of Mathematics 159 2004 217-250 Higher composition laws I A new view on Gauss composition and quadratic generalizations By Manjul Bhargava 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 composition 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 composition 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 X 2 X 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 X 2 X 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 X 2 X 2 cubes also .

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