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Erschienen in:
2016 | OriginalPaper | Buchkapitel
2. Mumford Curves
verfasst von : Werner Lütkebohmert
Erschienen in: Rigid Geometry of Curves and Their Jacobians
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Abstract
The incentive to create the theory of holomorphic functions over a non-Archimedean field was Tate’s elliptic curve. By means of rigid geometry one can explain Tate’s elliptic curve from the geometric point of view, whereas Tate originally formulated it in terms of function fields; cf. Sect. 2.1.
In the following sections we study Mumford’s generalization of Tate’s curve to curves of higher genus in the context of rigid geometry. We introduce discontinuous actions of certain subgroups \(\varGamma\) of \(\operatorname{PGL}(2,K)\) on the projective line in the style of Schottky. The structure of these groups \(\varGamma\) was found by Ihara; cf. Sect. 2.2.
Mumford curves will be introduced as orbit spaces \(\varGamma\backslash \varOmega\), where \(\varOmega\subset\mathbb{P}_{K}^{1}\) is the largest subdomain of \(\mathbb{P}_{K}^{1}\) on which \(\varGamma\) acts in an ordinary way. The construction of the quotient \(\varGamma\backslash\varOmega\) can be carried out in the framework of classical rigid geometry. Note that Mumford achieves much more general results in his ground braking article (Mumford in Compos. Math. 24:129–174, 1972) which deals exclusively with formal schemes. The concept here follows geometric constructions in order to explain the ideas behind Mumford’s construction.
Chapter 2 is somehow a counterpart of Riemann surfaces and their Jacobians. We provide the full picture of Mumford curves and their Jacobians which are rigid analytic tori. We show the duality theory of rigid analytic tori, Riemann’s period relations and, moreover, Riemann’s vanishing theorem.
Our approach is a refined version of the work of Drinfeld and Manin (J. Reine Angew. Math. 262/263:239–247, 1973) and Manin (Itogi Nauki Teh., Ser. Sovrem. Probl. Mat. 3:5–92, 1974), where they work over a \(p\)-adic field; i.e., a finite extension of \(\mathbb{Q}_{p}\). Here we consider a general non-Archimedean field as defined in Definition 1.1.1; notably we mention the work of Gerritzen (Math. Ann. 210:321–337, 1974; J. Reine Angew. Math. 297:21–34, 1978; Math. Ann. 196:323–346, 1972).