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Zbl 0609.14018
Jordan, Bruce W.; Livné, Ron A.
On the Néron model of Jacobians of Shimura curves. (English)
[J] Compos. Math. 60, 227-236 (1986). ISSN 0010-437X; ISSN 1570-5846/e

Let ${\cal B}$ be an indefinite quaternion algebra of discriminant Disc(${\cal B})>1$. Let $V\sb{{\cal B}}=V\sb{{\cal B}}/{\bbfQ}$ be the Shimura curve corresponding to ${\cal B}$. Fix a bad prime p of $V\sb{{\cal B}}$ (that is, $p\vert Disc({\cal B}))$, and denote by ${\cal J}/{\bbfZ}\sb p$ the Néron model of the Jacobian of $V\sb{{\cal B}}\otimes\sb{{\bbfQ}}{\bbfQ}p$, by ${\cal J}\sp 0\sb p$ the connected component of the special fiber ${\cal J}\sb p={\cal J}\times\sb{{\bbfZ}\sb p}{\bbfF}\sb p$ and by $\Phi ={\cal J}\sb p/{\cal J}\sp 0\sb p$ the group of connected components. \par In the paper under review, a formula for the order of $\Phi$, denoted $\vert \Phi \vert$ is obtained, and the structure theorem for ${\cal J}\sp 0\sb p/{\bbfF}\sb p$ is proved. Let $\hat {\cal B}$ be the rational definite quaternion algebra of discriminant Disc(${\cal B}/p)$ and let m($\hat {\cal B})$ be the mass of $\hat {\cal B}$ (m($\hat {\cal B})=12\sp{-1}\prod\sb{q\vert Disc \hat {\cal B}}(q-1)).\quad Let$ $B=B(p)$ be the Brandt matrix of degree $p$ for $\hat {\cal B}$. (Then $B\in M\sb h({\bbfZ})$ where h is the class number of $\hat {\cal B}$, and it has $p+1$ as its eigenvalue.) \par Theorem 1. Let $e\sb 2=\prod\sb{q\vert Disc {\cal B}}(1-(\frac{- 4}{q})),\quad e\sb 3=\prod\sb{q\vert Disc {\cal B}}(1-(\frac{- 3}{q})).\quad Then\vert \Phi \vert =((p+1)/m(\hat {\cal B})c(\hat {\cal B})2\sp{e\sb 2}3\sp{e\sb 3})\vert \prod\sp{h}\sb{i=2}(\lambda\sb i- (p+1))(\lambda\sb i+(p+1))\vert $ where c($\hat {\cal B})=8$ if Disc(${\cal B})=2$, c($\hat {\cal B})=3$ if Disc(${\cal B})=3$, c($\hat {\cal B})=1$ otherwise. Fix a maximal order $\hat {\cal M}\subset \hat {\cal B}$ and set $\Gamma\sb+=\{x\in (\hat {\cal M}\otimes {\bbfZ}[1/p])\sp{\times}\vert Norm(x)\in p\sp{2{\bbfZ}}\}/{\bbfZ}[1/p]\sp{\times}.$ \par Denote by $\Delta$ the Bruhat-Tits building of $SL\sb 2({\bbfQ}\sb p)$. $\Gamma\sb+$ acts on $\Delta$ with quotients of a finite oriented graph. Let $w\sb p$ denote an involution of $\Gamma\sb+\setminus \Delta$. Then $\Gamma\sb+\setminus \Delta$ is canonically identified with the dual graph $G=G({\cal M}\times {\bbfZ}\sb p/{\bbfZ}\sb p)$ of the special fiber ${\cal M}\sb{{\cal B}}\times {\bbfF}\sb p$, and Frobenius acts on G as $w\sb p$. Noting that all components of the special fiber (${\cal M}\sb B\times {\bbfZ}\sb p)\sb 0$ are rational so that the connected component ${\cal J}\sp 0\sb p$ is a torus, the structure theorem for ${\cal J}\sp 0\sb p/{\bbfF}\sb p$ is proved. \par Theorem 2. ${\cal J}\sp 0\sb p\approx H\sp 1((\Gamma\sb+\setminus \Delta),{\bbfZ})\otimes {\bbfG}\sb m$. The action of Frobenius is $w\sb p\otimes Frob\sb{{\bbfG}\sb m}$. - In particular, if $\ell =p$ is a prime, then the Tate module $Ta\sb{\ell}({\cal J}\sp 0\sb p)\approx H\sp 1((\Gamma\sb+\setminus \Delta),{\bbfZ}\sb{\ell})$ with Frobenius acting as $pw\sb p.$ \par These theorems are proved, on the basis of works of {\it M. Raynaud} [Publ. Math., Inst. Hautes Étud. Sci. 38, 27-76 (1970; Zbl 0207.516)] and of {\it P. Deligne} and {\it M. Rapoport} [Modular functions one Variable, II. Proc. Internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349, 143-316 (1973; Zbl 0281.14010)], first constructing a regular scheme ${\cal M}\sb{{\cal B}}\times {\bbfZ}\sb p\sim$ over ${\bbfZ}\sb p$, and then carrying out computations in linear algebra involving the Brandt matrix B.
[N.Yui]

MSC 2000:: *14H40 Jacobians
14E30 Minimal models
14H25 Arithmetic ground fields (curves)
11R52 Quaternion and other division algebras: arithmetic, zeta functions

Keywords: connected components; indefinite quaternion algebra; Shimura curve; Néron model of the Jacobian; Bruhat-Tits building

Citations: Zbl 0207.516; Zbl 0281.14010

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