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On the Néron model of Jacobians of Shimura curves. (English) Zbl 0609.14018

Let \({\mathcal B}\) be an indefinite quaternion algebra of discriminant Disc(\({\mathcal B})>1\). Let \(V_{{\mathcal B}}=V_{{\mathcal B}}/{\mathbb{Q}}\) be the Shimura curve corresponding to \({\mathcal B}\). Fix a bad prime p of \(V_{{\mathcal B}}\) (that is, \(p| Disc({\mathcal B}))\), and denote by \({\mathcal J}/{\mathbb{Z}}_ p\) the Néron model of the Jacobian of \(V_{{\mathcal B}}\otimes_{{\mathbb{Q}}}{\mathbb{Q}}p\), by \({\mathcal J}^ 0_ p\) the connected component of the special fiber \({\mathcal J}_ p={\mathcal J}\times_{{\mathbb{Z}}_ p}{\mathbb{F}}_ p\) and by \(\Phi ={\mathcal J}_ p/{\mathcal J}^ 0_ p\) the group of connected components.
In the paper under review, a formula for the order of \(\Phi\), denoted \(| \Phi |\) is obtained, and the structure theorem for \({\mathcal J}^ 0_ p/{\mathbb{F}}_ p\) is proved. Let \(\hat {\mathcal B}\) be the rational definite quaternion algebra of discriminant Disc(\({\mathcal B}/p)\) and let m(\(\hat {\mathcal B})\) be the mass of \(\hat {\mathcal B}\) (m(\(\hat {\mathcal B})=12^{-1}\prod_{q| Disc \hat {\mathcal B}}(q-1)).\quad Let\) \(B=B(p)\) be the Brandt matrix of degree \(p\) for \(\hat {\mathcal B}\). (Then \(B\in M_ h({\mathbb{Z}})\) where h is the class number of \(\hat {\mathcal B}\), and it has \(p+1\) as its eigenvalue.)
Theorem 1. Let \(e_ 2=\prod_{q| Disc {\mathcal B}}(1-(\frac{- 4}{q})),\quad e_ 3=\prod_{q| Disc {\mathcal B}}(1-(\frac{- 3}{q})).\quad Then| \Phi | =((p+1)/m(\hat {\mathcal B})c(\hat {\mathcal B})2^{e_ 2}3^{e_ 3})| \prod^{h}_{i=2}(\lambda_ i- (p+1))(\lambda_ i+(p+1))| \) where c(\(\hat {\mathcal B})=8\) if Disc(\({\mathcal B})=2\), c(\(\hat {\mathcal B})=3\) if Disc(\({\mathcal B})=3\), c(\(\hat {\mathcal B})=1\) otherwise. Fix a maximal order \(\hat {\mathcal M}\subset \hat {\mathcal B}\) and set \(\Gamma_+=\{x\in (\hat {\mathcal M}\otimes {\mathbb{Z}}[1/p])^{\times}| Norm(x)\in p^{2{\mathbb{Z}}}\}/{\mathbb{Z}}[1/p]^{\times}.\)
Denote by \(\Delta\) the Bruhat-Tits building of \(SL_ 2({\mathbb{Q}}_ p)\). \(\Gamma_+\) acts on \(\Delta\) with quotients of a finite oriented graph. Let \(w_ p\) denote an involution of \(\Gamma_+\setminus \Delta\). Then \(\Gamma_+\setminus \Delta\) is canonically identified with the dual graph \(G=G({\mathcal M}\times {\mathbb{Z}}_ p/{\mathbb{Z}}_ p)\) of the special fiber \({\mathcal M}_{{\mathcal B}}\times {\mathbb{F}}_ p\), and Frobenius acts on G as \(w_ p\). Noting that all components of the special fiber (\({\mathcal M}_ B\times {\mathbb{Z}}_ p)_ 0\) are rational so that the connected component \({\mathcal J}^ 0_ p\) is a torus, the structure theorem for \({\mathcal J}^ 0_ p/{\mathbb{F}}_ p\) is proved.
Theorem 2. \({\mathcal J}^ 0_ p\approx H^ 1((\Gamma_+\setminus \Delta),{\mathbb{Z}})\otimes {\mathbb{G}}_ m\). The action of Frobenius is \(w_ p\otimes Frob_{{\mathbb{G}}_ m}\). - In particular, if \(\ell =p\) is a prime, then the Tate module \(Ta_{\ell}({\mathcal J}^ 0_ p)\approx H^ 1((\Gamma_+\setminus \Delta),{\mathbb{Z}}_{\ell})\) with Frobenius acting as \(pw_ p.\)
These theorems are proved, on the basis of works of M. Raynaud [Publ. Math., Inst. Hautes Étud. Sci. 38, 27-76 (1970; Zbl 0207.516)] and of P. Deligne and 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 \({\mathcal M}_{{\mathcal B}}\times {\mathbb{Z}}_ p\sim\) over \({\mathbb{Z}}_ p\), and then carrying out computations in linear algebra involving the Brandt matrix B.
Reviewer: N.Yui

MSC:

14H40 Jacobians, Prym varieties
14E30 Minimal model program (Mori theory, extremal rays)
14H25 Arithmetic ground fields for curves
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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References:

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