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
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更高的复合定律 I：高斯复合和二次推广的新观点
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发表时间：
2004
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通讯作者：
M. Bhargava
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作者：
M. Bhargava
两个世纪前，在他著名的工作算术研究的1801年，高斯奠定了美丽的法律组成的整体二次型将发挥这样一个关键作用数论在几十年来。即使在两个世纪后的今天，这个合成定律仍然是理解和计算二次阶类群的主要工具之一。因此，很自然地要问，是否存在这种复合律的更高级的类似物，可以揭示其他代数数环和域的结构。这篇文章形成了一系列的四篇文章中的第一篇，我们的目的正是为了发展这种“更高的组成规律”。事实上，我们表明，高斯的法律组成只是一个至少有14个法律组成的同类产品的信息数环及其类组。本文开始，首先导出2×2×2立方整数上的一个一般合成律，并由此得到二元二次型上的高斯合成律，作为一个简单的特例，其形式类似于平面椭圆曲线上的群律。我们还从2× 2 × 2立方体上的这个合成律得到了另外四个新的合成律。这些合成定律定义在1）二元三次形式，2）二元二次形式对，3）四元交替2-形式对，和4）六元（六变量）交替3-形式上。更确切地说，高斯定理指出，给定判别式D的本原二元二次型的SL 2（Z）-等价类的集合具有固有的群结构。上面提到的其他五种形式的空间（包括2 × 2 × 2立方体空间）也具有Z上特殊线性群及其某些乘积的自然作用。我们证明，就像高斯的二元二次型空间，这些群作用有以下显着的性质。首先，这六个空间中的每一个对于相应的群作用只有一个多项式不变量，我们称之为判别式。发现该判别不变量仅取
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 2×2×2 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