Invisibility of Tate-Shafarevich groups in abelian surfaces (Q2929642)

Let \(E/\mathbb{Q}\) be an elliptic curve. The Galois cohomology group \(H^1(\mathbb{Q},E)\) is interpreted as equivalence classes of twists \(C\) of \(E\), known as the Weil-Châtalet group of \(E\), which are equipped with addition map \(C\times E \to C\), and the map \(\theta\): \(E \to C\) given by \(\theta(P) = p_0 + P\) for a fixed point \(p_0\in C\). Via \(\theta\) the curve \(C\) looks as if it is a coset of \(E\). Mazur suggested trying to visualize elements of the cohomology group as cosets of \(E\) inside some larger abelian variety \(A\). Given an inclusion \(E \to A\), the visible subgroup of \(H^1(\mathbb{Q},E)\) is defined to be the kernel of \(H^1(\mathbb{Q},E) \to H^1(\mathbb{Q},A)\). The visibility dimension of an element \(\xi \in H^1(\mathbb{Q},E)\) is the least dimension of an abelian variety \(A\). This notion of visibility dimension is also considered for the Tate-Shafarevich group \(\text Ш(E/\mathbb{Q})\), the subgroup of \(H^1(\mathbb{Q},E)\) consisting of \(C\) that are locally elliptic curves. \textit{B. Mazur} [Asian J. Math. 3, No. 1, 221--232 (1999; Zbl 0958.11043)] proved that all elements of order \(3\) in \(\text Ш(E/\mathbb{Q})\) are visible in an abelian surface. It is unlikely that every element of \(\text Ш(E/\mathbb{Q})\) would be visible in an abelian surface, but it does not seem that we had examples of \(E\) for which some elements of \(\text Ш(E/\mathbb{Q})\) have visibility dimension \(3\). As an application the author of the paper under review introduces examples of elliptic curves for which some elements of \(\text Ш(E/\mathbb{Q})[n]\) where \(n=6\) or \(7\) are not visible in an abelian surface, but visible in an abelian three-fold.NEWLINENEWLINELet us introduce the main result of the paper.NEWLINENEWLINEGiven \(E\), let \( E_1,\dots,E_t\) be representatives for the isogeny classes of elliptic curves such that \(E[n]\cong E_k[n]\) as Galois modules. Suppose that \(E[\ell]\) is irreducible for all primes \(\ell\) dividing \(n\), and \(\mathrm{rank}\, E(\mathbb{Q})=0\), and we further assume some technical conditions if \(\mathrm{rank}\, E_k(\mathbb{Q})>0\) for some \(k\); see the paper for detail conditions. Then, the number of elements in \(H^1(\mathbb{Q},E)\) of order \(n\) that are visible in an abelian surface is at most \(\sum_{k=1}^t v_k\) where \(v_k\) is the number of elements of order \(n\) in \(E_k(\mathbb{Q})/nE_k(\mathbb{Q})\).NEWLINENEWLINE For \(n=6\) or \(7\) the author introduces examples of elliptic curves \(E\), and for each \(E\), the complete list of \(E'\) such that \(E[n]\cong E'[n]\). Using Magma he verifies that the size of \(\text Ш(E/\mathbb{Q})[n]\) is too big for this list of \(E'\), and hence, by the main result stated above, some elements in \(\text Ш(E/\mathbb{Q})[n]\) are verified to be not visible in an abelian surface.