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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

##### MSC:
 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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