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


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
Full Text: Numdam EuDML


[1] L. Breen et J.P. Labesse : Variétés de Shimura et fonctions L . Publications Mathématiques de l’Université de Paris VII, No. 6. · Zbl 0517.00003
[2] H. Carayol : Sur la mauvaise réduction des courbes de Shimura . Comptes-Rendus de l’Académie des Sciences, t. 296, Série I (1983) p. 557. · Zbl 0599.14019
[3] H. Carayol : Sur les représentations l-adiques attachées aux formes modulaires de Hilbert . Comptes-Rendus de l’Académie des Sciences, t. 296, Série I (1983) p. 629. (Les deux notes précitées contiennent quelques erreurs de signes, dont j’espère qu’elles sont corrigées dans le présent article). · Zbl 0537.10018
[4] C. Chevalley : Deux théorèmes d’arithmétique . J. Math. Soc. Japan 3 (1951) p. 36. · Zbl 0044.03001
[5] P. Deligne : Travaux de Shimura . Séminiare Bourbaki, février 1971 (première version). · Zbl 0286.14007
[6] P. Deligne : Travaux de Shimura . Séminaire Bourbaki, février 1971 (seconde version). In: Lecture Notes in Mathematics, No. 244, p. 123, Springer-Verlag (1971). · Zbl 0225.14007
[7] P. Deligne : Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques . Proceedings of Symposia in Pure Mathematics, 33 (1979), Part 2, pp. 247-290, Amer. Math. Soc., Providence (1979). · Zbl 0437.14012
[8] P. Deligne et D. Mumford : On the irreducibility of the space of curves of a given genus . Publ. Math., IHES, No. 36 (1968). · Zbl 0181.48803
[9] V.G. Drinfel’D : Elliptic Modules Math., URSS, Sbornik, Vol. 23, No. 4 (1974). · Zbl 0321.14014
[10] J. Giraud : Modules de variétés abéliennes et variétés de Shimura, exposé dans [Br] . · Zbl 0538.14030
[11] N.M. Katz et B. Mazur : Arithmetic Moduli of Elliptic Curves . Princeton Univ. Press: Princeton (1985). · Zbl 0576.14026
[12] M. Knesser : Hasse principle for H1 of simply connected group , in Algebraic Groups and Discontinuous subgroups . Proc. Symp. in Pure Math. Vol. IX, Amer. Math. Soc., Providence R.I. (1966) pp. 159-163. · Zbl 0259.20041
[13] M. Lichtenbaum : Curves over discrete valuation rings . Amer. J. of Maths. 90 (1968) 380. · Zbl 0194.22101
[14] R.P. Langlands : Modular forms and l-adic representations, Modular forms of one variable II . Lecture Notes Math., No. 349, Springer-Verlag (1973) 361-500. · Zbl 0279.14007
[15] W. Messing : The crystals associated to Barsotti-Tate groups, with applications to abelian schemes . Lecture Notes in Math. No. 264, Springer-Verlag (1972). · Zbl 0243.14013
[16] Y. Morita : Reduction mod p of Shimura curves . Hokkaido Math. J., 10 (1981) 209-238. · Zbl 0506.14021
[17] D. Mumford : Geometric Invariant Theory . Springer-Verlag (1965). · Zbl 0147.39304
[18] I Šafarevich : Lectures on minimal models . Tata Inst. Lecture Notes, Bombay (1966).
[19] J.P. Serre : Local class field theory . In: Cassels-Fröhlich; Algebraic Number Theory . Washington, Thomson Book Co. (1967).
[20] G. Shimura : On canonical models of arithmetic quotients of bounded symmetric domains . Annals of Math. 91 (1970) 144-222. · Zbl 0237.14009
[21] J. Tate : p-divisible groups . In: Proceedings of a Conference on Local Fields, Nuffic summer School at Driebergen, Springer-Verlag (1967). · Zbl 0157.27601
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