  Let E be a one-parameter family of elliptic curves over a number field. It is
natural to expect the average root number of the curves in the family to be
zero. All known counterexamples to this folk conjecture occur for families
obeying a certain degeneracy condition. 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 non-degenerate family E has average root
number 0, provided that two classical arithmetical conjectures hold for two
homogeneous polynomials with integral coefficients constructed explicitly in
terms of E.
  The first such conjecture -- commonly associated with Chowla -- asserts the
equidistribution of the parity of the number of primes dividing the integers
represented by a polynomial. We prove the conjecture for homogeneous
polynomials of degree 3.
  The second conjecture used states that any non-constant homogeneous
polynomial yields to a square-free sieve. We sharpen the existing bounds on the
known cases by a sieve refinement and a new approach combining height
functions, sphere packings and sieve methods.
