One-Class Genera of Positive Quadratic Forms in Seven Variables
One-Class Genera of Positive Quadratic Forms in Seven Variables
复制标题
七变量正二次型的一类生成
DOI：
10.1112/plms/s3-48.1.175
复制
发表时间：
1984
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
设f是一个正定二次型，系数为整数，变量个数为n，c（f）是它的类数，即f所属的亏格中的类数。我在[4]中证明了对于所有n^ 11的f，c（f）> 1，并且从那时起试图确定所有c（f）= 1和3^ n^ 10的f; n= 1的情况是平凡的，我的方法不足以满足n= 2。对于n= 3的情况，见[7]，需要特殊处理，包括对复杂结果的重压缩。给出结果的最简单的方法是忽略非本原/，只列出每个单类属的一个代表。现在，如果我们通过从每个倒易对中剔除一个形式来进一步缩短列表，则在简单的情况下，其中的形式数分别减少到2或1，n= 9或10，在[10]中处理，而对于n= 8，则减少到22，见[11]。本文将证明，当n= 7时，只有87个原始单纲属，其中之一是平方和x1 2+...+。x72，其余的构成43个互逆对。我们将看到，这一陈述可以从下面的定理推导出来。
Let/be a positive-definite quadratic form, with integer coefficients, in any number n of variables, and let c (f) be its class-number, that is, the number of classes in the genus to which/belongs. I proved in [4] that c (f)> 1 for all/with n^ 11, and have since sought to determine all the/with c (f)= 1 and 3^ n^ 10; the case in which n= 1 is trivial and my methods do not suffice for n= 2. The case where n= 3, see [7], needed special treatment, including heavy condensation of the complicated result. The simplest way to present the result would be to ignore imprimitive/and list just one representative of each one-class genus. Now if we further shorten the list by rejecting one form from each reciprocal pair, the number of forms in it decreases to 2 or 1 in the easy cases, n= 9 or 10 respectively, dealt with in [10], and to 22 for n= 8, see [11]. In this paper it will be shown that for n= 7 there are just 87 primitive one-class genera; one of them is the sum of squares x1 2+...+ x7 2, and the others constitute 43 reciprocal pairs. We shall see that this statement can be deduced from the following theorem.