Elliptic curves with all quartic twists of the same root number (Q2075293)

For a number field \(K\) and an elliptic curve, \(E/K\), defined over \(K\), we can associate an \(L\)-function, \(L(E/K,s)\), known as its Hasse-Weil \(L\)-function. This function is defined for \(\mathrm{Re}(s)>3/2\) and the Hasse-Weil conjecture asserts that it has an analytic continuation to the entire complex plane. The Hasse-Weil conjecture also states that this \(L\)-function satisfies a functional equation under \(s \longleftrightarrow 2-s\) with a factor of \(\pm 1\) on one side. This factor of \(\pm 1\) is called the (global) root number of the curve, \(w(E/K)\). The functional equation implies that \(w(E/K)=(-1)^{\mathrm{ord}_{s=1} L(E/K,s)}\), so, under the Birch Swinnerton-Dyer conjecture, the root number tells us the parity of the rank of Mordell-Weil group, \(E(K)\). In this paper, the authors complete an investigation of when the root number stays constant for all twists of an elliptic curve. There are four types of twists: quadratic, cubic, quartic and sextic. All but the third of these (the quartic twists) have been considered in previous work. See [\textit{T. Dokchitser} and \textit{V. Dokchitser}, Acta Arith. 137, No. 2, 193--197 (2009; Zbl 1275.11097)] and [\textit{D. Byeon} and \textit{N. Kim}, J. Number Theory 136, 22--27 (2014; Zbl 1284.11089)]. Here the authors address the remaining case, that of the quartic twists. Only elliptic curves with \(j\)-invariant \(1728\) defined by the equation \(y^{2}=y^{2}+ax\) can have quartic twists and these twists are given by \(E_{D}/K: y^{2}=x^{3}+aDx\) where \(D \in K^{\times}/ \left( K^{\times} \right)^{4}\). Their Theorem~1.1 states that the root numbers of all quartic twists are constant if and only if \(\sqrt{-1} \in K\). Furthermore, the root number is always \(+1\) if \(\sqrt{-1} \in K\). If \(\sqrt{-1} \not\in K\), then there are infinitely many \(D\) such that \(w \left( E_{D}/K \right)=+1\) and infinitely many \(D\) such that \(w \left( E_{D}/K \right)=-1\). \textit{K. Česnavičius} [J. Reine Angew. Math. 719, 45--73 (2016; Zbl 1388.11032)] already proved that if \(\sqrt{-1} \in K\), then any elliptic curve with \(j\)-invariant \(1728\) over \(K\) has root number \(+1\). Here the authors use the structure of the Galois group \(\mathrm{Gal} \left( L/K_{v} \right)\), where \(K_{v}\) is a local field with respect to a place \(v|2\) and \(L=K_{v}(E[3])\), to establish the behaviour when \(\sqrt{-1} \not\in K\).