Transformations of a Quadratic Form Which Do Not Increase the Class‐Number
Transformations of a Quadratic Form Which Do Not Increase the Class‐Number
复制标题
不增加类数的二次形式的变换
DOI：
10.1112/plms/s3-12.1.577
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发表时间：
1962
影响因子：
1.8
通讯作者：
G. Watson
中科院分区：
文献类型：
--
作者：
G. Watson
研究具有整系数的二次型的类数，即亏格中的类数，有时可以通过考虑具有更简单性质的另一种形式来简化。例如，众所周知，倒数形式具有相同的类号，因为等价形式具有等价的倒数。有关另一个示例，请参见(1)。在这里，Jones考虑了一个三元形式/=f{xx2，x3)，其中p是素数，pf2a；并且证明了这样的三元形式的类数不小于g=p1f(px1，x2，x3)。这很有用，因为g的判别式在数值上小于i的判别式。这两个结果都包含在本文的定理1中。Magnus在(3)中证明，如果n^35，每个正的w元形式亏格至少包含两个类。在一篇正在准备中的论文中，我将利用这篇论文的结果证明，具有n^11的正旋量亏格总是至少包含两类。常数11是最好的可能。引入旋量亏格主要是因为它不会使论证变得复杂；有关它的描述，请参阅CH。第(4)项中的7项。本文件是单独发表的，因为这些论点与上述即将发表的论点不同，而且这些结果本身似乎有些意思，但对于积极形式而言，它是无限期有效的。然而，不幸的是，就不确定形式而言，它们似乎没有什么有趣的应用。
THE investigation of the class-number, that is the number of classes in the genus, of a quadratic form with integral coefficients can sometimes be simplified by considering another form with simpler properties. For example, it is well known that reciprocal forms have the same classnumber, because equivalent forms have equivalent reciprocals. For another example, see (1). Here Jones considered a ternary form/= f {xv x2, x3) with the property/= axx 2 (modp) identically in the variables, where p is prime, pf2a; and showed that the class-number of such an/is not less than that of g= p~ 1f (px1, x2, x3). This is useful because the discriminant of g is numerically less than that of/. Both these results are included in Theorem 1 of this paper. Magnus showed in (3) that every positive genus of w-ary forms contains at least two classes if n^ 35. I shall show, in a paper under preparation, using the results of this paper, that a positive spinor-genus with n^ 11 always contains at least two classes. The constant 11 is best possible. The spinor-genus is introduced mainly because it does not seriously complicate the argument; for an account of it see Ch. 7 of (4). The present paper is published separately because the arguments are different from those of the forthcoming one referred to above, and because the results, which seem to be of some interest in themselves, are valid for indefinite as for positive forms. Unfortunately, however, they seem to have no interesting applications as far as indefinite forms are concerned.