## 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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