Let \(R\) be a discrete valuation ring with fraction field \(K\) and let \(A_K\) be an abelian variety with Néron model \(A_R\) and component group \(\phi_{A_R}\). Let \(A^t_K\) be the dual abelian variety with Néron model \(A^t_R\) and component group \(\phi_{A^t_R}\). A. Grothendieck [“Groupes de monodromie en géométrie algébrique” (SGA 7I), Lect. Notes Math. 288 (1972; Zbl 0237.00013)] constructed a pairing \(\phi_{A_R}\times\phi_{A^t_R}\rightarrow\mathbb{Q}/\mathbb{Z}\) that represents the obstruction to extending the Poincaré bundle to a biextension of \(A_R\times A^t_R\) by \(\mathbb{G}_m\), which conjecturally is a perfect pairing. While this holds in a variety of situations, notably the case of semi-stable reduction, the authors’ main result is that it may fail for non-perfect residue field.
The authors’ main device is the following: Let \(K'/K\) be a finite Galois extension of degree \(n\) with purely inseparable residue extension. Let \(X_K=\text{Res}_{K'/K} A_{K'}\), where \(A_{K'} = A_K\times_K K'\) is the base change of \(A_K\) to \(K'\). Let \(X^t_K =\text{Res}_{K'/K} A^t_{K'}\). Assuming that Grothendieck’s conjecture is true for \(A_{K'}\) the authors obtain information on that pairing for \(X_K\) and \(X^t_K\) (theorem 2.1). In particular, they show (corollary 2.2) that, if \(\text{char}(k)=p\), \(K'/K\) is unramified, \(n = p^r\) and the \(p\)-part of \(\phi_{A^t_{R'}}\) is not trivial, Grothendieck’s conjecture fails for \(X_K\) and \(X^t_K\).