Idempotent relations and the conjecture of Birch and Swinnerton-Dyer
Let \(A\) be an abelian variety defined over a number field \(K\), and let \(L(A,s)\) denote its global \(L\)-function. If \(L(A,s)\) has an analytic continuation around \(s=1\) and if the Shafarevich-Tate group is finite, then the leading coefficient of the Taylor expansion of the \(L\)-function is conjecturally given by the so-called constant \(C(A/K)\) of Birch and Swinnerton-Dyer. The author proves various identities concerning the constants \(C(A/K')\) as \(K'\) varies over the subfields \(K'\) of a finite Galois extension \(L/K\) of \(K\) with Galois group \(G\). These identities arise from idempotent relations in the group ring \(\mathbb Q[G]\) in the following way: Let \(\varepsilon_H = \frac{1}{| H| } \sum_{\sigma\in H}\,\sigma\) denote the idempotent attached to the subgroup \(H\). The author shows that under the assumption that the Shafarevich-Tate group is finite a relation \(\sum_H n_H \varepsilon_H = 0\) between the idempotents implies the relation \(\prod_H \,C(A/L^H)^{n_H} = 1.\) In the special case of elliptic curves this was conjectured by \textit{H. Park} [``Idempotent relations and the conjecture of Birch and Swinnerton-Dyer'', in: Algebra and Topology (Taejon, 1990), 97--125 (1990; Zbl 0736.11029)]. The author then considers the more general situation of a twisted group ring \(\text{End}_L(A)[G]\), introduces twisted idempotent relations and proves similar results involving twists of abelian varieties.
scientific article; zbMATH DE number 2001048
scientific article