On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic \(p>0\)
Let \(A\) be an Abelian variety over a global field \(K\) of positive characteristic \(p\) and assume that \(A\) has good reduction everywhere. Let \(\text Ш (A/K)\) be the Tate-Shafarevich group of \(A\). The author proves that the finiteness of the \(p\)-primary component \(\text Ш (A/K) (p)\) implies the finiteness of \(\text Ш (A/K)\) and the validity of the Birch and Swinnerton-Dyer conjecture for \(A\) [For constant \(A\) this result is due to \textit{J. S. Milne}, Invent. Math. 6, 91--105 (1968; Zbl 0159.224); see also Ann. Math. (2) 102, 517--533 (1975; Zbl 0343.14005)]. The author notices that a result of \textit{P. Schneider} [Math. Ann. 260, 495--510 (1982; Zbl 0509.14022)] allows to replace the finiteness assumption for \(\text Ш (A/K)(P)\) by the finiteness assumption for the \(\ell\)-primary component \(\text Ш (A/K) (\ell)\) for any prime \(\ell\).
scientific article; zbMATH DE number 149674
scientific article