三元二次型与虚二次域类数

发布时间：2018-03-24 23:38

本文选题：三元　切入点：二次型　出处：《南京大学》2016年博士论文

【摘要】：给定非负整数n和正定整系数三元二次型Q(x,y,z),我们称方程Q{x,y,z) = n整数解的个数为Q表n的表示数,记为RQ(n)。本文,我们对三元二次型：x2+y2 + z2, x2+y2 + 2z2, x2 + y2 + 3z2, x2 + y2 + 4z2, x2+y2 + 5z2, x2 + y2 + 6z2, x2 + y2 + 8z2, x2+2y2 + 3z2, x2 + 2y2 + 4z2, x2 + 2y2 + 6z2, x2 + 3y2 + 6z2, x2 + 4y2+4z2, x2 + 4y2 + 8z2, 2x2+2y2+3z2, 2x2+3y2+3z2, x2+2y2+2z2, x2+3y2+3z2, x2 + 5y2+5z2, x2+6y2 + 6z2, 2x2+3y2+6z2, x2+y2+2z2+yz, x2+y2+7z2, x2+y2+11z2, x2+y2 + 13z2, x2+y2 + 19z2, x2+3y2 + 5z2逐一进行讨论,揭示它们的表示数与对应虚二次域类数的关系。其中最后五个二次型,因为它们的类数大于1,我们需要考虑其genus内所有类的二次型表示数,进而建立它们的线性组合与对应虚二次域类数的关系。下面我们例举我们得到的若干结果。假设p是一个除去有限个例外值的奇素数,令Q=x2+y2 + 2z2 + yz,则我们有：类似的,令Q1=x2+3y2 + 5z2, Q2 = x2 + 2y2 + 8z2-2yz,则我们有这里h(d)表示虚二次域Q((?)d)的类数。我们在文中还会给出一些“对偶”的结果,比如对Q的表示数,我们有需要提到的是,二次型x2+y2+3z2的情形是由孙智宏教授提出的一个猜想,这个猜想在最近被郭-彭-秦[3]证明。本文我们将揭示上述这一现象广泛地存在于一般表示数与类数之间。我们首先主要对裴定一得到解析公式的二十个对应尖形式空间为零的对角型正定整系数三元二次型进行讨论,得到类似的若干关系式。进一步的,我们对更多对应尖形式空间不为零的三元二次型(且未必为对角型)加以讨论,建立其解析公式,得到类似上述的关系式,并给出证明。我们还特别对x2+py2+qz2型(p,q是奇素数)的三元二次型进行了深入的研究,通过计算模形式尖点处的值,结合genus中其他代表元,建立其表示数与虚二次域类数的公式。在本文的最后一章,我们还对Cooper和Lam提出的关于表示数的一系列猜想做了进一步的探讨,证明了b=1,c=21时猜想成立。精确的说,我们证明了RQ(n2)= 4H(1,21,n)。这里其中ep表示p模n的指数。对于猜想中其他没有解决的某些情形,比如b=3,c=10时,由于对应二次型的类数为1,且对应爱森斯坦级数空间的基与b=1,c=21有类似的形式,故我们有希望用相同的方法来给予证实。不过,本文我们没有给出具体的证明。
[Abstract]:Given a non-negative integer n and a positive definite integral coefficient of the quaternary quadratic form Q ~ (x) ~ y ~ (z1), we call the number of integer solutions of the equation Q {x ~ (y) ~ z = n the representation number of Q table n, which is denoted as RQ _ n _ n. In this paper, We have three quadratic forms x2y2z2, x2y22z2, x2y23z2x2y24z2x2y25z2, x2y25z2x2y26z2x2y28z2x2 2y2 3z2x2 2y2 4z2x2 2y2 6z2x2 3y2 6z2x2 4y2 4z2x2 4y2 8z2 2x2 2y2 3z2 2x2 3y2 3z2x2 2y2 2z2x2#en11# 3z2x2#en12# 5z2x2#en13# 6z2x2 3y2 6z2x2y2. 2z2 yz, x2y27z2, x2y211z2, x2y213z2x2y219z2x2 3y2 5z2 are discussed one by one. The relation between their representation numbers and the number of classes corresponding to virtual quadratic fields is revealed. The last five quadratic forms, because the number of classes is greater than 1, we need to consider the quadratic representation numbers of all classes in their genus. Then we establish the relation between their linear combination and the class number of the corresponding virtual quadratic field. We give some results below. Suppose p is an odd prime number with the exception of finite number, let Q=x2 y 2 2z2 y z, then we have the following:. Let Q1=x2 3y2 5z2, Q2 = x2 2y2 8z2-2yz. then we have here hmd) to denote the virtual quadratic field QG? We will also give some results of "duality" in this paper, for example, for the representation number of Q, we need to mention that the case of quadratic type x2y2 3z2 is a conjecture put forward by Professor Sun Zhihong. This conjecture has recently been proved by Guo Peng-Qin [3]. In this paper, we will reveal that this phenomenon widely exists between the general representation number and the class number. First, we obtain twenty corresponding tips of the analytic formula for Pei Dingyi. The ternary quadratic form of diagonal positive definite integral coefficient with zero formal space is discussed. Some similar relations are obtained. Further, we discuss more ternary quadratic forms (and not necessarily diagonal forms) corresponding to the apical form space, and establish their analytical formulas, and obtain the relations similar to the above. It is also proved that the ternary quadratic form of x2 py2 qz2 type qz2 is an odd prime number. By calculating the value at the cusp of the modular form and combining with other representative elements in genus, we also give a further study on the ternary quadratic form of x2 py2 qz2 type. In the last chapter of this paper, we further discuss a series of conjectures about representation numbers put forward by Cooper and Lam, and prove that the conjecture BX 1C = 21:00 holds. We prove that RQN _ 2N _ (2) = 4H ~ (1) H ~ ((-1)) ~ (21) N ~ (-1), where EP denotes the exponent of p-module n. For some other unsolved cases in the conjecture, such as BX _ 3C = 10:00, the number of classes corresponding to the quadratic form is 1, and the base of the corresponding space of the Eisenstein series has a similar form to b1c21. So we hope to prove it in the same way. However, we do not give any concrete proof in this paper.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O156

【相似文献】