On the behaviour of root numbers in families of elliptic curves

Abstract

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.

65 pages; major changes

References in corpus (2)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves

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• The parity problem for reducible cubic forms
• On the variation of the root number in families of elliptic curves
• Density of rational points on a family of del Pezzo surfaces of degree one
• On the rank of the fibres of rational elliptic surfaces
• Constant root number on integer fibres of elliptic surfaces
• Power-free values, large deviations, and integer points on irrational curves
• On the rank of the fibers of elliptic K3 surfaces