Constructing non-trivial elements of the Shafarevich-Tate group of an abelian variety over a number field (Q1946715)

Let \(A\) be an abelian variety defined over a number field \(K\) and, for any positive integer \(n\), let \({\text Ш}(A/K)[n]\) be the \(n\)-torsion of the Tate-Shafarevich group of \(A\) over \(K\). The order of the full group \({\text Ш}(A/K)\) appears in the Birch and Swinnerton-Dyer conjecture, hence it is an important (but still quite mysterious) invariant of the variety. The paper provides a way of constructing elements in \({\text Ш}(A/K)[n]\) (or to make them ``visible'') as images of rational points of another abelian variety \(B\) whose \(n\)-torsion is isomorphic to \(A[n]\), following (and generalizing) results of \textit{J. E. Cremona} and \textit{B. Mazur} [Exp. Math. 9, No.~1, 13--28 (2000; Zbl 0972.11049)]. The main theorem can be summarized as follows: Let \(A\) and \(B\) be abelian varieties defined over a number field \(K\) such that \(A[n]\simeq B[n]\). Let \(n\) be an odd integer such that (i) for any prime \(p\) dividing \(n\), the maximal ramification index of a prime of \(K\) lying over \(p\) is \(< p-1\); (ii) \(\displaystyle{\gcd\left( n, N\cdot |A(K)_{\text{tor}}|\cdot \prod_v c_{B,v}\cdot c_{A,v} \right) =1 }\) (where \(N\) is the product of the residue characteristics of all primes of bad reduction for either \(A\) or \(B\) and \(c_{B,v}\), \(c_{A,v}\) are the Tamagawa numbers). Then there is a map \[ \varphi : B(K)/nB(K) \rightarrow {\text Ш}(A/K)[n] \] which is injective if \(\text{rk}\,(A(F))=0\) and an isomorphism if \({\text Ш}(B/K)[n]=0\). The proof is based on the description of the Tate-Shafarevich group via flat cohomology of the Neron model \(\mathcal{A}\) of \(A\) over \(\text{Spec}\,\mathcal{O}_K\) (introduced in [\textit{B. Mazur}, Invent. Math. 18, 183--266 (1972; Zbl 0245.14015)]) and on the exact sequence \[ 0\rightarrow A(K)/nA(K) \rightarrow H^1(\text{Spec}\,\mathcal{O}_K,\mathcal{A}[n]) \rightarrow {\text Ш}(A/K)[n] \rightarrow 0 \] arising from Kummer theory (plus some hypotheses on \(n\)). The isomorphism \(A[n]\simeq B[n]\) links the exact sequence above with the analogous one for \(B\), providing the map \(\varphi\). Application for the case \(A=E\) (an optimal elliptic curve), \(K=\mathbb{Q}\) and \(n=5\) are provided in the introduction of the paper.