On Grothendieck’s pairing of component groups in the semistable reduction case. (English) Zbl 0872.14037

Werner, Annette

J. Reine Angew. Math. 486, 207-217 (1997).

Let \(K\) be a discretely valued field, and let \(A_K\) be an abelian variety over \(K\). We denote by \(A\) its Néron model and by \(A'\) the Néron model of the dual abelian variety \(A_K'\). Let \(\varphi_A\) and \(\varphi_{A'}\) be the groups of connected components of the special fibres of \(A\) and \(A'\). Then there is a canonical pairing \((*): \varphi_A \times \varphi_{A'} \to\mathbb{Q}/ \mathbb{Z}\), which appears as obstruction of extending the Poincaré biextension over \(A_K \times A_K'\) to \(A \times A'\). A conjecture of Grothendieck states that this pairing is perfect. If \(A\) has semistable reduction, we can use the monodromy pairing to define another pairing \((**): \varphi_A \times \varphi_{A'} \to\mathbb{Q}/ \mathbb{Z}\), which is easily seen to be perfect. Following a request of A. Grothendieck [in Sém. Géométrie Algébr. 1967-1969, SGA 7, I, No. 9, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006); 11.4] we show that both pairings are compatible, which proves Grothendieck’s conjecture in the semistable reduction case. We use the rigid geometry of the Raynaud extensions associated to \(A_K\) and \(A_K'\) and the theory of formal Néron models of rigid analytic groups.

Reviewer: A.Werner (Münster)

MSC:

14K15	Arithmetic ground fields for abelian varieties
14G20	Local ground fields in algebraic geometry
12J10	Valued fields

Keywords:

valued field; abelian variety; semistable reduction; rigid geometry; formal Néron models