On the behaviour of root numbers in families of elliptic curves

Details On the behaviour of root numbers in families of elliptic curves On the behaviour of root numbers in families of elliptic curves

2004-08-10

arxiv.org/abs/math/0408141v3

Mathematics

Number Theory

65 pages; major changes

Scientific paper

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.

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