Grothendieck's pairing for Jacobians and base change
In the following, let \(K\) be the field of fractions of a complete discrete valuation ring with separably closed residue field \(k\) with \(\mathrm{char}(k)=p>0\), and let \(A\) be an abelian variety over \(K\). We denote \(A'=\mathrm{Pic}^0_A\). By Grothendieck's theory, there exists a natural pairing \(\langle \cdot, \cdot \rangle_A: \Phi_{A,K} \times \Phi_{A',K} \rightarrow \mathbb{Q}/\mathbb{Z}\) where \(\Phi_{A,K}\) and \(\Phi_{A',K}\) denote the component groups of the Néron models of \(A\) and \(A'\), respectively. Due to work of Bégueri, Bertapelle, Bosch, McCallum, Lorenzini and Werner, the pairing is known to be perfect in many cases. The pairing is conjectured to be perfect if \(k\) is perfect. There are counter examples which show that the pairing is not perfect in general. The main result of this article extends an earlier result of Bosch and Lorenzini. Let \(C\) be a smooth geometrically connected proper curve of genus \(\geq 1\) over \(K\). We denote the Jacobian of \(C\) by \(J=J(C)\). Let \(\varphi:J \rightarrow J'\) be the canonical principal polarization of \(J\). It is convenient to identify \(J\) with \(J'\) via the morphism \(- \varphi\). This identification has a continuation to Néron models, and hence induces an isomorphism \(\Phi_{J,K} \rightarrow \Phi_{J',K}\). As a consequence, Grothendieck's pairing induces a symmetric pairing \(\langle \cdot, \cdot \rangle_{J,K}:\Phi_{J,K} \times \Phi_{J,K} \rightarrow \mathbb{Q}/\mathbb{Z}\) on the component group \(\Phi_{J,K}\) of the Néron model of \(J(C)\). The main result of Lorenzini reads as follows. Theorem. Assume that \(C\) has a proper flat regular integral model \(\mathcal{C}\) such that the irreducible components of the reduction \(\mathcal{C}_k\) are smooth, \(\mathcal{C}_k^{\mathrm{red}}\) has normal crossings and the multiplicities of the irreducible components have no proper common factor. If there exist two distinct irreducible components of \(\mathcal{C}_k\) with strictly positive intersection number and coprime multiplicities, then the pairing \(\langle \cdot, \cdot \rangle_{J,K}\) is perfect. The proof of the above theorem makes use of an earlier result of \textit{S.~Bosch} and \textit{D.~Lorenzini}, Invent. Math. 148, No. 2, 353--396 (2002; Zbl 1061.14042) Th.4.6] which says that, under the theorem's assumptions, for the pairing \(\langle \cdot , \cdot \rangle_{J,K}\) to be perfect it suffices that \(C(K) \not= \emptyset\). The idea of proof is that this situation can be achieved over a carefully chosen base extension \(L\) of \(K\). Consider the morphism \(\gamma:\Phi_{J,K} \rightarrow \Phi_{J,L}\) which is induced by base change. Let \(e_{L/K}\) denote the ramification index of the extension \(L/K\). By the formula \(\big\langle \gamma(x), \gamma(y) \big\rangle_{J,L} = e_{L/K} \cdot \big\langle x,y \big\rangle_{J,K}\) for all \(x,y \in \Phi_{J,K}\), one can relate the perfectness of the pairing \(\langle \cdot , \cdot \rangle_{J,K}\) to the one of \(\langle \cdot , \cdot \rangle_{J,L}\), provided that \(e_{L/K}\) is invertible modulo the group orders of \(\Phi_{J,K}\) and \(\Phi_{J,L}\). In fact, Lorenzini proves that one can choose \(L=K \big( \pi^{1/n} \big)\), where \(\pi\) denotes a uniformizer of \(K\) and where \(n\) is a large enough prime. Here the assumption on the existence of intersecting components with coprime multiplicities comes into play.
scientific article; zbMATH DE number 5285724
scientific article