Constant root number on integer fibres of elliptic surfaces

Abstract

Rizzo showed that the family W t : y 2 = x 3 + tx 2 − (t + 3)x + 1, a well-known example of Washington, is such that W (W t ) = −1 for all t ∈ Z [Riz03]. In this paper, we fully determinate the non-isotrivial 1-parameter families of elliptic curves E t with coefficients (in parameter t) of small degree such that the root number is the same for all E t with t ∈ Z.(1) Chowla's conjecture predicts the behavior of λ(n), the parity of the number of prime factors of n, when n varies through the values of a polynomial:The conjecture is known to be true for deg f ≤ 1, which makes Corollary 3 unconditional relatively to this conjecture for families with at most one (linear) factor of the discriminant ∆ not dividing the c 4 -invariant.(2) The Squarefree conjecture estimates the proportion of squarefree values of a polynomial:This conjecture is known to hold when every factors of f has degree at most 3. Thus the only case where Corollary 3 is conditional relatively to this conjecture is the case where the c 4 -coefficient of a family is irreducible over Q(T ). Families with coefficients of bounded degreesLet E be a family of elliptic curves given for t ∈ Q by the Weierstrass equationwhere deg a i ≤ 2 for i = 2, 4, 6. Suppose that there are no factor of ∆ that divides the c 4 -invariant (or in other words there is no place of multiplicative reduction: the family is potentially parity-biaised ), then Bettin, David and Delauney prove in recent work [BDD18, Theorem 7 and 8] that there are essentially 6 different classes of non-isotrivial such famillies, namely: