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Drinfeld modular curves. (English) Zbl 0607.14020
Lecture Notes in Mathematics, 1231. Berlin etc.: Springer-Verlag. xiv, 107 pp. DM 23.00 (1986).
This monograph introduces the reader to the function field analogue of the theory of elliptic modular curves. Beginning with a review of Drinfeld modules, lattices, and partial zeta functions the author quickly proceeds to a study of Drinfeld’s upper half-plane, its quotient by an arithmetic subgroup, and the compactification of the quotient by adjoining finitely many cusps. This leads naturally to a study of the expansions at the cusps of certain modular forms which may be thought of as function field analogues of the Fricke functions and the discriminant function \(\Delta\). The author deduces from this a formula for the genus of the modular curves associated to maximal arithmetic subgroups, and shows that the cuspidal divisor class group of such a curve is finite (the function field analogue of the Manin-Drinfeld theorem).
There is also a brief discussion of algebraic modular forms, and of modular curves attached to congruence subgroups. Finally, the author introduces the Hecke operators, both as correspondences on the set of 2-lattices with level structure, and as endomorphisms of the Jacobian of the modular curve. In the latter connection there is a discussion of the occurrence of elliptic curves as factors of the “new” part of the Jacobian \(J_ 0\). The book concludes with a list of open questions.
Reviewer: S. Kamienny

14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
11G09 Drinfel’d modules; higher-dimensional motives, etc.
14H25 Arithmetic ground fields for curves
11F52 Modular forms associated to Drinfel’d modules
14G25 Global ground fields in algebraic geometry
11-02 Research exposition (monographs, survey articles) pertaining to number theory
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
11R58 Arithmetic theory of algebraic function fields
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