On the behaviour of root numbers in families of elliptic curves
On the behaviour of root numbers in families of elliptic curves
复制标题
关于椭圆曲线族中根数的行为
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
H. Helfgott
中科院分区：
文献类型：
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作者：
H. Helfgott
设E是环Q上的单参数椭圆曲线族。我们证明了平均根数为零的一大类相当一般类型的椭圆曲线的家庭。此外，我们证明了任何族E在Q（t）上至少有一个乘法约化点，其平均根数为0，只要两个经典的算术定理对两个用E显式构造的多项式成立.在Q（t）上没有乘法约化的任何族E中的根数的行为被证明是相当规则和非随机的;在这种情况下，我们给出了平均根数的表达式。
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root number in this case.