On Néron class groups of abelian varieties (Q2879868)

Let \(F\) be a global field, \(S\) a nonempty finite set of primes (containing all the infinite primes in the number field case) and \({\mathcal O}_{F,S}\) the corresponding ring of \(S\)-integers. Let \(A\) be an abelian variety over \(F\), and \(\mathcal A\) the Néron model of \(A\) over \(U=\text{Spec}\, {\mathcal O}_{F,S}\) with connected component \({\mathcal A}^\circ\). Write \(D^1(U, {\mathcal A})\) for the subgroup of elements of the étale cohomology group \(H^1(U, {\mathcal A})\) that are mapped to 0 by the restriction maps to completions \(H^1(U, {\mathcal A})\to H^1(F_v, {\mathcal A})\) for all \(v\in S\), and similarly for \({\mathcal A}^\circ\). Finally, denote by \(C_{A, F, S}\) (resp. \(C^1_{A, F, S}\)) the kernel (resp. cokernel) of the natural map \(D^1(U, {\mathcal A}^\circ)\to D^1(U, {\mathcal A})\).NEWLINENEWLINEThe main result of the paper constructs, under assumption of the finiteness of the Tate-Shafarevich group of \(A\), a perfect pairing of finite groups \(C_{A, F, S}\times C^1_{B, F, S}\to{\mathbb Q}/{\mathbb Z}\), where \(B\) is the dual abelian variety of \(A\). This is achieved by first constructing a perfect pairing \(D^1(U, {\mathcal A}^\circ)\times D^1(U, {\mathcal B})\to{\mathbb Q}/{\mathbb Z}\) following the method of [\textit{J. S. Milne}, Arithmetic duality theorems. 2nd ed. Charleston, SC: BookSurge, LLC (2006; Zbl 1127.14001)], section II.5 and [\textit{D. Harari} and \textit{T. Szamuely}, J. Reine Angew. Math. 578, 93--128 (2005; Zbl 1088.14012)], and then combining it with the usual Cassels-Tate duality for \(A\).NEWLINENEWLINEThe main interest of this result comes from the interpretation of the groups \(C_{A, F, S}\) and \(C^1_{B, F, S}\). The first group can be identified with the cokernel of the natural reduction map \(A(F)\to\bigoplus_{v\in U}\Phi_v(A)(k(v))\), where \(\Phi_v(A)\) is the group scheme of connected components of the special fibre of the Néron model of \(A_{F_v}\). Grothendieck constructed a non-degenerate pairing between \(\Phi_v(A)(\overline{k(v)})\) and \(\Phi_v(B)(\overline{k(v)})\) which induces a nondegenerate pairing between \(\bigoplus_{v\in U}\Phi_v(A)({k(v)})\) and \(\prod_{v\in U}H^1(k(v),\Phi_v(B))\). The author identifies \(C^1_{B, F, S}\) with an explicitly described subgroup of the latter product, and shows that it is the exact orthogonal of the image of the map \(A(F)\to\bigoplus_{v\in U}\Phi_v(A)(k(v))\) under the above pairing.NEWLINENEWLINEThe author also remarks that \(C_{A, F, S}\) can be defined and calculated similarly for any connected smooth commutative \(F\)-group scheme that admits a Néron model over \(U\). In the case of the multiplicative group \({\mathbb G}_m\) one recovers the \(S\)-ideal class group of \(F\).