On the cuspidal divisor class group of a Drinfeld modular curve
Let \(K\), \(K_\infty\), \(A\), \(C\) and \(\Gamma\) be respectively a global field \(K\) of positive characteristic, its completion \(K_\infty\) at a fixed place \(\infty\), the ring \(A\) of elements of \(K\) regular outside \(\infty\), the completion \(C\) of an algebraic closure of \(K_\infty\) and an arithmetic subgroup \(\Gamma\) of \(GL_2 (K)\), i.e. a subgroup commensurable with \(GL_2 (A)\). Set \(\Omega= C-K_\infty\). This is a Drinfeld upper half-plane, \(\Gamma\) acts by fractional linear transformations on \(\Omega\) and the rigid analytic space \(M_\Gamma= \Gamma\setminus \Omega\) is indeed an affine curve over \(C\); this is a Drinfeld modular curve. These curves, for various \(\Gamma\), are the substitutes in positive characteristic of the classical modular curves. Let \(\overline{M}_\Gamma\) be the canonical completion of \(M_\Gamma\). The author studies divisors on \(\overline{M}_\Gamma\) with support in the set of cusps, i.e. in \(\overline{M}_\Gamma- M_\Gamma\) (cuspidal divisors). To use analytic methods [as in \textit{E.-U. Gekeler} and \textit{M. Reversat}, J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)], in \S 2, the author generalizes the theory of theta functions developed in (loc. cit.) to ``degenerate parameters''. One knows that the period lattice (in the sense of Manin-Drinfeld) of \(\overline{M}_\Gamma\) is described by a set of harmonic cochains or a set of theta functions (loc. cit.). The author adds to that interpretation the description of the links between new theta functions, cuspidal divisors and harmonic cochains (\S 3). In \S 4, where \(\Gamma\) is assumed to be a congruence subgroup, the author studies, with the help of the preceding methods, the canonical map between the group \({\mathcal C}(\Gamma)\) generated in the Jacobian \({\mathcal J}_\Gamma\) of \(\overline{M}_\Gamma\) by the cusps (the group \({\mathcal C}(\Gamma)\) is finite here), and the group \(\Phi_\infty (\Gamma)\) of connected components of the Néron model of \({\mathcal J}_\Gamma\) at \(\infty\). Finally (\S 5), in the case of \(A= \mathbb{F}_q [T]\) and for a Hecke congruence subgroup, the author obtains more information about his new theta functions and the (eventually non-empty) kernel of the map \({\mathcal C}(\Gamma)\to \Phi_\infty(\Gamma)\).
scientific article; zbMATH DE number 1139214