## 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

### Citations:

Zbl 0207.516; Zbl 0281.14010
Full Text:

### References:

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