Galois actions on Néron models of Jacobians

Halle, Lars H.

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 853-903.

Soit $X$ une courbe lisse définie sur le corps des fractions $K$ d’un anneau de valuation discrète $R$. Nous étudions une filtration naturelle sur la fibre spéciale du modèle de Néron de la Jacobienne de $X$ par des sous-schémas en groupes fermés unipotents. Nous démontrons que les sauts de cette filtration ne dépendent que du type de la fibre spéciale du modèle minimal régulier à croisements normaux stricts de $X$ sur $R$. En particulier, les sauts sont indépendants de la caractéristique résiduelle. Ensuite, nous obtenons des informations plus précises sur les sauts, et nous les calculons pour chaque type de fibre possible pour les courbes de genre 1 et 2.

Let $X$ be a smooth curve defined over the fraction field $K$ of a complete discrete valuation ring $R$. We study a natural filtration of the special fiber of the Néron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $R$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber types for curves of genus $1$ and $2$.

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