本文選題：三元　切入點(diǎn)：二次型　出處：《南京大學(xué)》2016年博士論文

【摘要】：給定非負(fù)整數(shù)n和正定整系數(shù)三元二次型Q(x,y,z),我們稱方程Q{x,y,z) = n整數(shù)解的個(gè)數(shù)為Q表n的表示數(shù),記為RQ(n)。本文,我們對(duì)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逐一進(jìn)行討論,揭示它們的表示數(shù)與對(duì)應(yīng)虛二次域類數(shù)的關(guān)系。其中最后五個(gè)二次型,因?yàn)樗鼈兊念悢?shù)大于1,我們需要考慮其genus內(nèi)所有類的二次型表示數(shù),進(jìn)而建立它們的線性組合與對(duì)應(yīng)虛二次域類數(shù)的關(guān)系。下面我們例舉我們得到的若干結(jié)果。假設(shè)p是一個(gè)除去有限個(gè)例外值的奇素?cái)?shù),令Q=x2+y2 + 2z2 + yz,則我們有：類似的,令Q1=x2+3y2 + 5z2, Q2 = x2 + 2y2 + 8z2-2yz,則我們有這里h(d)表示虛二次域Q((?)d)的類數(shù)。我們?cè)谖闹羞�會(huì)給出一些“對(duì)偶”的結(jié)果,比如對(duì)Q的表示數(shù),我們有需要提到的是,二次型x2+y2+3z2的情形是由孫智宏教授提出的一個(gè)猜想,這個(gè)猜想在最近被郭-彭-秦[3]證明。本文我們將揭示上述這一現(xiàn)象廣泛地存在于一般表示數(shù)與類數(shù)之間。我們首先主要對(duì)裴定一得到解析公式的二十個(gè)對(duì)應(yīng)尖形式空間為零的對(duì)角型正定整系數(shù)三元二次型進(jìn)行討論,得到類似的若干關(guān)系式。進(jìn)一步的,我們對(duì)更多對(duì)應(yīng)尖形式空間不為零的三元二次型(且未必為對(duì)角型)加以討論,建立其解析公式,得到類似上述的關(guān)系式,并給出證明。我們還特別對(duì)x2+py2+qz2型(p,q是奇素?cái)?shù))的三元二次型進(jìn)行了深入的研究,通過計(jì)算模形式尖點(diǎn)處的值,結(jié)合genus中其他代表元,建立其表示數(shù)與虛二次域類數(shù)的公式。在本文的最后一章,我們還對(duì)Cooper和Lam提出的關(guān)于表示數(shù)的一系列猜想做了進(jìn)一步的探討,證明了b=1,c=21時(shí)猜想成立。精確的說,我們證明了RQ(n2)= 4H(1,21,n)。這里其中ep表示p模n的指數(shù)。對(duì)于猜想中其他沒有解決的某些情形,比如b=3,c=10時(shí),由于對(duì)應(yīng)二次型的類數(shù)為1,且對(duì)應(yīng)愛森斯坦級(jí)數(shù)空間的基與b=1,c=21有類似的形式,故我們有希望用相同的方法來給予證實(shí)。不過,本文我們沒有給出具體的證明。
[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.
【學(xué)位授予單位】：南京大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O156

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