zbMATH — the first resource for mathematics

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

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