Drinfeld modular curves / Ernst-Ulrich Gekeler

his 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 Δ. 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). ... (Zentralblatt)