Shimura curves. (Les courbes de Shimura.) (French) Zbl 0723.14018

van der Put, M.

Sémin. Théor. Nombres Bordx., Sér. II 1, No. 1, 89-102 (1989).

This article is a nice review of recent and not quite so recent results on Shimura curves: their complex uniformization and modular interpretation, the theorem of Cherednik-Drinfeld, which says that Shimura curves are, for suitable primes \(p\), Mumford curves and describe explicitly their \(p\)-adic uniformization, and finally the theorem of Ribet, who found a surprising relation between the Jacobians of a \(p\)-adic Shimura curve and certain \(q\)-adic modular curves, for different primes \(p\) and \(q\).

Reviewer: F.Herrlich (Bochum)

MSC:

14G35	Modular and Shimura varieties
14H25	Arithmetic ground fields for curves
14K15	Arithmetic ground fields for abelian varieties

Keywords:

complex uniformization; Mumford curves; Jacobians of a \(p\)-adic Shimura curve; \(q\)-adic modular curves

References:

[1]	Drinfeld, V.G.Coverings of p-adic symmetric regions, Funct. Analysis and its appl.10 (1976) n°2, 107-115. · Zbl 0346.14010
[2]	Jordan, B.W. & Livné, R.A.Local Diophantine Properties of Shimura curves. Math. Ann.270, 235-248 (198). · Zbl 0536.14018
[3]	Katsura, T. & Oort, F.Families of supersingular abelian surfaces. Compositio Math.62, 107-167 (1987). · Zbl 0636.14017
[4]	Ribet, K.A.On modular representations of Gal(Q/Q) arising from modular forms. (Preprint Sept. 19, 1988).
[5]	Ribet, K.A.Bimodules and abelian surfaces. (Preprint, August 10, 1988, PAM-423, Berkeley). · Zbl 0742.11033