Elliptic curves with a lower bound on 2-Selmer ranks of quadratic twists (Q2391651)

Let \(K\) be a number field, \(E\) an elliptic curve defined over \(K\) and, for any quadratic extension \(F/K\), let \(E^F\) be the associated quadratic twist of \(E\). Let \(\text{Sel}_2(E/K)\) be the 2 Selmer group, i.e., \[ \text{Sel}_2(E/K):=\text{Ker}\left\{ H^1(K,E[2])\longrightarrow \prod_v H^1(K_v,E[2])/\text{Im}\,\kappa_v\right\} \] (where \(v\) runs through all the primes of \(K\), the map is the product of natural restrictions and \(\kappa_v\) is the local Kummer map). The paper deals with the \textit{2-Selmer rank} \[ d_2(E^F/K):=\dim_{\mathbb{F}_2}\text{Sel}_2(E^F/K)-\dim_{\mathbb{F}_2}E(K)[2] \] as \(F\) varies through all the quadratic twists of \(E\). It is known (see [\textit{T. Dokchitser} and \textit{V. Dokchitser}, J. Reine Angew. Math. 658, 39--64 (2011; Zbl 1314.11041)]) that \(d_2(E^F/K)\equiv d_2(E/K) \pmod 2\) for any \(E^F\) and it was conjectured that any value \(r \equiv d_2(E/K) \pmod 2\) could be obtained by infinitely many quadratic twists of \(E\) (see [\textit{B. Mazur} and \textit{K. Rubin}, Invent. Math. 181, No. 3, 541--575 (2010; Zbl 1227.11075)]). The present paper provides counterexamples to this conjecture using the family of elliptic curves \[ E_n : y^2+xy=x^3-128n^2x^2-48n^2x-4n^2\quad\text{(}n\in \mathbb{N}\text{)} \] defined over \(\mathbb{Q}\). For any number field \(K\) with \([K:\mathbb{Q}]=r_1+2r_2\) and \(r_2>0\), such that \(1+256n^2\not\in(K^*)^2\) the curves \(E_n\) verify {\parindent=6mm \begin{itemize}\item[1.] \(|E_n(K)[2]|=2\) and there exist an isogeny of degree 4 defined over \(K(E[2])\) but not over \(K\); \item[2.] \(E_n\) has multiplicative reduction at primes \(\mathfrak{p}|2n\) and Kodaira symbol \(I_{2v_{\mathfrak{p}}(2n)}\) at \(\mathfrak{p}\), while the isogenous curve \(E_n':=E_n/E_n(K)[2]\) has Kodaira symbol \(I_{4v_{\mathfrak{p}}(2n)}\) at \(\mathfrak{p}\); \item[3.] the curves \(E_n\) have distinct \(j\)-invariants, hence are not isomorphic over an algebraic closure \(\overline{K}\) of \(K\). \end{itemize}} The author uses these properties, local computations for the images of Kummer maps and exact sequences involving different Selmer groups to show that \(d_2(E^F/K)\geq r_2\) for any quadratic twist (i.e., ``small'' \(r \equiv d_2(E/K) \pmod{2}\) are not obtainable). It is worth remarking that, since for any \(K\) there are infinitely many \(n\) such that \(1+256n^2\not\in(K^*)^2\,\), this paper provides infinitely many (non-isomorphic by 3.) counterexamples to the conjecture mentioned above.