On the product of twists of rank two and a conjecture of Larsen (Q1039638)

Let \(E\) be an elliptic curve defined over \(\mathbb Q\). Let \(W(E)\) be its root number. The parity conjecture asserts that \(W(E)=1\) if and only if the rank of \(E(\mathbb Q)\) is even. Let \(d\) be an element of \(\mathbb Q^*/\mathbb Q^*{^2}\) and \(E_d\) be the quadratic twist of \(E\) by \(\sqrt{d}\). Let us denote by \(W(E,d)\) the root number of \(E_d\). In case \(d\) and \(d'\) are fundamental discriminants, relatively prime to the conductor of \(E\), one has the formula \[ W(E,dd')=W(E,d)W(E,d')W(E). \] Suppose that the rank of \(E(\mathbb Q)\) is even. If the ranks of \(E_d(\mathbb Q)\) and \(E_{d'}(\mathbb Q)\) are even at least two, then assuming the parity conjecture, the above formula implies that the rank of \(E_{dd'}(\mathbb Q)\) is also even. Generically, according to a conjecture of Goldfeld, the rank of \(E_{dd'}(\mathbb Q)\) should be equal to zero. In this paper, the author gives examples of elliptic curves \(E\) over \(\mathbb Q\) and infinite parametric families of pairs of integers \((d,d')\) as above, such that, assuming the parity conjecture, the ranks of \(E_d(\mathbb Q)\), \(E_{d'}(\mathbb Q)\) and \(E_{dd'}(\mathbb Q)\) are all even at least two. As an application, the author shows examples where a conjecture of Larsen holds.