The Parity Problem for Reducible Cubic Forms

This conjecture can be traced to Chowla ([2], p. 96); it is closely related to the Bunyakovsky/Schinzel conjecture on primes represented by irreducible polynomials. The one-variable analogue of (1.2) is classical for deg f = 1 and quite hopeless for deg f > 1. We know (1.2) itself when deg f ≤ 2. (The main ideas of the proof go back to de la Vallée-Poussin ([5], [6]); see [10], §3.3, for an exposition.) The problem of proving (1.2) when deg f ≥ 3 has remained open until now: sieving is forestalled by the parity problem ([16]), which Chowla’s conjecture may be said to embody in its pure form. We prove (1.2) for f reducible of degree 3. In a companion paper ([11]), we prove (1.2) for f irreducible of degree 3. Part of the importance of Chowla’s conjecture resides in its applications to problems of parity outside analytic number theory. Knowing that (1.2) holds for deg f = 3 allows us to conclude that in certain one-parameter families of elliptic curves the root number W (E) = ±1 averages to 0 ([8], Proposition 5.6). In §5, we will show that the two-parameter family y = x(x + a)(x + b) has average root number 0 as well. In the process, we will see that, for some f , (1.2) is robust under certain twists by characters to variable moduli.