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On the bad reduction of Shimura curves. (Sur la mauvaise réduction des courbes de Shimura.) (French) Zbl 0607.14021

Suppose \(B\) is a quaternionic algebra over \(F\), \(F/{\mathbb Q}\) a totally real number field of degree \(d\), such that \(B\otimes_ F{\mathbb R}\simeq M_ 2({\mathbb R})\times H^{d-1}\). Set \(G:=\text{Res}_{F/{\mathbb Q}}B^{\times}\) and consider the compact Shimura curves \(M_ K(G,X)/F\), with \(X={\mathbb C}-{\mathbb R}\) and \(K\subset G({\mathbb A}_ F)\) an open and compact subgroup. Fix a place \({\mathfrak p}\) of \(F\), where \(B\) is split, and consider the subgroups \(K=K^ n_{{\mathfrak p}}\times H\), with \(K^ n_{{\mathfrak p}}=\{x\in \text{GL}_ 2({\mathfrak O}_{{\mathfrak p}})\mid x\equiv \text{Id} \bmod {\mathfrak p^ n}\}\) and \(H\) open and compact in \(\prod_{\mu \neq {\mathfrak p}}(B\otimes F_{\mu})^{\times}\), and the Shimura curves \(M_{n,H}=M_{K^ n_{{\mathfrak p}}\times H}\). Let \(\nu: G\to T,\) \(T=\text{Res}_{F/{\mathbb Q}}G_ m\), be the reduced norm. Then the curves \(M_{n,H}\) are defined over the finite \(F\)-schemes \({\mathcal M}_{n,\nu (H)}\) being the “Shimura varieties” relative to \(T\). Further one can define for each \((n,H)\) a scheme \(E_{n,H}\to M_{0,H}\) as well as \(L_{n,\nu (H)}\to {\mathcal M}_{n,\nu (H)}\), such that one obtains a pairing \(E_ n\times_{M_{0,H}}E_ n\to \nu^*L_ n\), \(E_ n=\lim_{\leftarrow} E_{n,H}\) being a substitute for the \(p^ n\)-torsion part of a universal elliptic curve and \(L_ n=\lim_{\leftarrow} L_{n,\nu (H)}.\)
In this paper it is shown that for \(H\) small enough all the curves \(M_{n,H}\) have projective models over \({\mathfrak O}_{{\mathfrak p}}\) and that the ”algebraic objects” \({\mathcal M}_{n,\nu (H)}\), \(E_ n\), \(L_ n\), \(M_{n,H}\to {\mathcal M}_{n,\nu (H)}\) and \(E_ n\times_{M_{0,H}}E_ n\to \nu^*L_ n\) can be compatibly lifted. If \((M_{0,H})_{(x)}\) denotes the completion of the strict localization in a geometric point x of the projective model \(M_{0,H}\), then \(E_{\infty}| (M_{O,H})_{(x)}\) represents the deformations of \(E_{\infty | x}\), and therefore the \({\mathfrak O}_{{\mathfrak p}}\)-morphism \(M_{O,H}\to {\mathcal M}_{0,\nu (H)}\) must be smooth. As a consequence one obtains the good reduction of \(M_{0,H}\) in \({\mathfrak p}\). (An analogous result holds for \({\mathcal M}_{0,\nu (H)}.)\) But the \({\mathfrak O}_{{\mathfrak p}}\)-morphism \(M_{n,H}\to {\mathcal M}_{n,\nu (H)}\) is only smooth outside the supersingular points and therefore \(M_{n,H}\) can not have good reduction because of the existence of supersingular points.
Reviewer: Maria Heep

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11G15 Complex multiplication and moduli of abelian varieties
14H25 Arithmetic ground fields for curves
14K22 Complex multiplication and abelian varieties
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References:

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