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Contributions to Shimura curves. (English) Zbl 1279.11062

Cojocaru, Alina-Carmen (ed.) et al., WIN–women in numbers: Research directions in number theory. Papers evolved out of the WIN workshop, Banff, Canada, November 2–7, 2008. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences (ISBN 978-0-8218-5226-2/hbk). Fields Institute Communications 60, 15-33 (2011).
From the introduction: In this paper, we provide an outline of joint results obtained with M. Alsina [Quaternion orders, quadratic forms, and Shimura curves. Providence, RI: AMS (2004; Zbl 1073.11040)], J. Guàrdia [J. Théor. Nombres Bordx. 17, No. 1, 57–67 (2005; Zbl 1093.11042)] and A. Travesa [Acta Arith. 126, No. 4, 315–339 (2007; Zbl 1158.11031); Pure Appl. Math. Q. 4, No. 4, 1107–1132 (2008; Zbl 1193.11037)] concerning the computation of explicit developments for automorphic functions of the canonical models and that of equations defining CM points on Shimura curves. To simplify the exposition, we formulate our results in the simplest non-modular case, given by the rational quaternion algebra of discriminant \(D=6\).
Section headings are as follows: 1. Basic concepts on rational quaternion algebras. 2. The Shimura curves \(X_D\). 3. Canonical models. 4. Local expansions. 5. Fake elliptic curves. 6. Suggestions for further research.
For the entire collection see [Zbl 1210.11005].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
11F03 Modular and automorphic functions
33E30 Other functions coming from differential, difference and integral equations
32N15 Automorphic functions in symmetric domains
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