CONSTRUCTING NEW NON-CONGRUENT NUMBERS BY GRAPH THEORY
CONSTRUCTING NEW NON-CONGRUENT NUMBERS BY GRAPH THEORY
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DOI：
10.1142/9789812770134_0002
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发表时间：
2007-07
期刊：
影响因子：
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通讯作者：
K. Feng;Y. Xue
中科院分区：
文献类型：
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作者：
K. Feng;Y. Xue
本文综述了近年来由椭圆曲线的传统数论加上代数图论的一个结果所得到的新的非全等数列的结果。更确切地说，我们从以下两个事实出发：（1）一个无平方正整数是一个非全等数当且仅当椭圆曲线En：y2= x3-n2 x的有理点群En（n）的秩为零。(2)若椭圆曲线E_n及其对偶曲线的2-塞尔默群S_n和S_n有极小尺寸|SN| = 1，则rank（En（n））= 0。塞尔默群可以由椭圆曲线Enand的齐次空间的局部可解性条件的数据来确定。下一步是将数据组织成精心构造的图，使得塞尔默群具有最小值当且仅当图具有特定的“奇”性质。利用代数图论中的一个结果，一个奇图可以用图的拉普拉斯矩阵的秩来描述。𝔽因此，通过计算矩阵的秩，我们可以确定所有的Sn和有最小值，在这种情况下，是一个非全等数。𝔽在解释了这种方法和有关概念后，我们给出了用这种方法得到的新的非全等数列的结果，并举例说明。
This paper is a survey on recent results of new series of non-congruent numbers which can be acquired by traditional arithmetic theory of elliptic curves plus a result from algebraic graph theory. More precisely, we start from the following two facts: (1) A square-free positive integeris a non-congruent number if and only if the rank of the group En(ℚ) of rational points of the elliptic curve En: y2= x3- n2x is zero. (2) If the 2-Selmer groups Snandof the elliptic curve Enand its dual curvehave minimal sizes |Sn| = 1 and, then rank(En(ℚ)) = 0. Selmer groups can be determined by the data of locally solvability conditions of the homogenous spaces of elliptic curves Enand. The next step is to organize the data into carefully constructed graphs so that the Selmer groups have minimum if and only if the graphs have specific "odd" property. By a result in algebraic graph theory, an odd graph can be described by the rank of the Laplace matrix of the graph over 𝔽2. Thus, by computing the rank of a certain matrix over 𝔽2, we can determine allsuch that Snandhave minimum, in which caseis a non-congruent number. After explaining this method and related concepts, we describe the results on new series of non-congruent numbers obtained in this way and illustrate them by examples.