On the cuspidal divisor class group of a Drinfeld modular curve
type: Item
instance of: scholarly article
title: On the cuspidal divisor class group of a Drinfeld modular curve
zbMATH Open document ID: 0895.11024
author: Ernst-Ulrich Gekeler
published in: Documenta Mathematica
publication date: 1998-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/230962
full work available at URL: 14.html
review text: 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…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1139214
zbMATH Keywords: class group
zbMATH Keywords: Drinfeld upper half-plane
zbMATH Keywords: Drinfeld modular curve
zbMATH Keywords: theta functions
zbMATH Keywords: cuspidal divisors
zbMATH Keywords: harmonic cochains
zbMATH Keywords: congruence subgroup
reviewed by: Marc Reversat
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1139214