0 M ay 2 00 3 Root Numbers and the Parity Problem Harald

Let E be a one-parameter family of elliptic curves over a number field K. 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. More precisely: given a homogeneous polynomial f ∈ Z[x, y], it is believed that μ(f(x, y)) averages to zero. This conjecture can be said to represent the parity problem in its pure form, while covering the same notional ground as the Bunyakovsky-Schinzel and Hardy-Littlewood conjectures taken together. For deg f = 1 and deg f = 2, Chowla’s conjecture is essentially equivalent to the prime number theorem. For deg f > 2, the conjecture has been unproven up to now; the traditional approaches by means of analysis and sieve theory fail. We prove the conjecture for deg f = 3. There remains to state the second arithmetical conjecture referred to previously. It is believed that any non-constant homogeneous polynomial f ∈ Z[x, y] 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.