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Groups of components of Néron models of Jacobians. (English) Zbl 0737.14008
Let \(A\) be a complete discrete valuation ring with quotient field \(K\), and algebraically closed residue field \(k\). Let \(X\) be a complete non- singular curve over \(K\), \({\mathcal X}\) be the minimal model of \(X\) over \(A\), and \(J\) be the Jacobian variety of \(X\). Moreover let \({\mathcal J}\) be the Néron model of \(J\). In this paper, the author considers the order \(\varphi\) of the étale group scheme consisting of the components of the special fibre of \({\mathcal J}\), and he gives an estimation of \(\varphi\) using the difference between the Euler-Poincaré characteristics of the special fibre \({\mathcal X}_s\) and the generic fibre \({\mathcal X}_\eta\) of \({\mathcal X}\), under the condition that the gcd of the multiplicities of the irreducible components of \({\mathcal X}_s\) is equal to one. In the proof, he uses a result of M. Raynaud [Publ. Math., Inst. Hautes Étud. Sci. 38, 27–76 (1970; Zbl 0207.51602)]. Moreover, he gives an improvement for a result of H. W. Lenstra jun. and F. Oort [J. Pure Appl. Algebra 36, 281–298 (1985; Zbl 0557.14022)].

MSC:
14H40 Jacobians, Prym varieties
14H25 Arithmetic ground fields for curves
14E30 Minimal model program (Mori theory, extremal rays)
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