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Rational points of the group of components of a Néron model. (English) Zbl 0934.14029

Structure and properties of groups of components of Néron models are investigated for Jacobians of algebraic curves, algebraic tori and abelian varieties over a discrete valuation field \( K\). Let \( A_K \) be an abelian variety over \(K.\) Let \(A\) be the Néron model of \(A_K\) over the ring of integers \({\mathfrak O}_K \) of \(K\) and \( A_k \) its special fibre over the residue field \( k \) of \({\mathfrak O}_K.\) Let \({\varphi}_A\) be the group of components of \(A.\) From results of S. Lang [Am. J. Math. 78, 555-563 (1956; Zbl 0073.37901)], S. Bosch, W. Lütkebohmert and M. Raynaud [“Néron models” (1990; Zbl 0705.14001)] and X. Xarles [J. Reine Angew. Math. 437, 167-179 (1993; Zbl 0764.14009)] follows that
(i) if \(K\) is henselian, then the canonical homomorphism \(A_K(K) \rightarrow A_k(k) \) is surjective;
(ii) if \(k\) is finite or if the identity component \( A^{0}_k\) of \(A_k\) is an extension of a unipotent group by a split torus with \(k\) perfect, then the homomorphism \( A_k(k) \rightarrow {\varphi}_A(k)\) is surjective (lemma 2.1).
Some interesting results about the group of components of a Néron model were obtained by D. Lorenzini [J. Reine Angew. Math. 445, 109-160 (1993; Zbl 0781.14029)].
The main results of the paper under review:
(I) Components groups of Jacobians. Let \(X\) be a proper flat regular curve over \({\mathfrak O}_K \) whose generic fibre is geometrically irreducible. Irreducible components of the special fibre \(X_k\) generate a free \(\mathbb Z\)-module \(\mathbb Z^I\). The authors define two homomorphisms of the \(\mathbb Z\)-modules which are essential for the computing of \({\varphi}_A(k).\) Let \( d_i \) be the multiplicity of \({\Gamma}_i\) in \(X_k\), let \(e_i\) be its geometric (in \(X \times_k k^s\) where \(k^s\) is a separable closure of \(k\)) multiplicity. Let \(r_i\) be the number of irreducible components of \((\Gamma_i)_{k^s}.\) For two divisors \(V_1,V_2\) on \(X\), such that at least one is vertical let \(\langle V_1,V_2\rangle\) be their intersection number. Then the homomorphism \(\alpha\) is defined by \[ \alpha(V)=\sum_i r_i^{-1}e_i^{-1}\langle V,\Gamma_i\rangle_{k}\Gamma_i \] for any \(V\subset{\mathbb Z}^I\). The homomorphism \(\beta\) is defined by \(\beta({\Gamma}_i)= r_{i}d_{i}e_{i}.\) Let \(d=\text{gcd}\{d_i\mid i\in I\}\) and \(d'=\text{gcd} \{r_id_i\mid i\in I\}\). Then (if \(\text{Gal}(k^{s}/k)\) is procyclic) the authors give an exact sequence \[ 0 \rightarrow\text{Ker }\beta/\text{Im }\alpha \rightarrow {\varphi}_A(k) \rightarrow (qd \mathbb Z)/d'\mathbb Z \rightarrow 0 \] with \(q=1\) if \(d'|g-1\) (\(g\) is the genus of \(X\)) and \(q=2\) otherwise (theorem 1.17).
(II) Algebraic tori. Let \(T_K\) be a torus over \(K\) with multiplicative reduction and let \(T\) be its Néron model over \({\mathfrak O}_K\). Let \(\varphi_T\) be the associated component group. For the Galois group \(G\) of the extension \(k^{s}/k\) the group of characters \(X\) of \(T_K\) is a \(G\)-module. Let \(X_G\) be the biggest \(\mathbb Z\)-free quotient of \(X\) which is fixed by \(G\) and let \(T_{G,K}\) be the torus with group of characters \(X_G\) which is the biggest subtorus which is split over \(K.\) Then the injection \(T_{G,K} \hookrightarrow T_K \) and the associated morphism of Néron models \(T_G \rightarrow T\) induce a monomorphism of component groups \(\varphi_{T_G} \hookrightarrow \varphi_T \) and an isomorphism \(\varphi_{T_G}(k) \simeq \varphi_{T}(k) \) between groups of \(k\)-rational points. Furthermore, the canonical map \(T_K(K) \rightarrow \varphi_{T}(k) \) is surjective, as the same is true for the split torus \(T_{G,K}\) (proposition 3.2).
(III) Abelian varieties with semi-stable reduction. Let \(A_K\) be an abelian variety with split semi-stable reduction; i.e., the authors assume that the identity component \(A_k^0\) of the special fibre of the Néron model \(A\) of \(A_K\) is the extension of an abelian variety by a split algebraic torus. Then: (i)The component group \(\varphi_A\) is constant (also valid if \(K\) is not necessary complete). (ii) The canonical map \( A_K(K) \rightarrow \varphi_{A}(k) \) is surjective (proposition 4.3).
Examples are given to show how these results may be used in calculation of component groups of Jacobians of curves.

MSC:

14K15 Arithmetic ground fields for abelian varieties
14H40 Jacobians, Prym varieties
14G05 Rational points
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