On the cuspidal divisor class group of a Drinfeld modular curve

Abstract

The theory of theta functions for arithmetic groups $\Gamma$ that act on the Drinfeld upper half-plane is extended to allow degenerate parameters. This is used to investigate the cuspidal divisor class groups of Drinfeld modular curves. These groups are finite for congruence subgroups $\Gamma$ and may be described through the corresponding quotients of the Bruhat-Tits tree by $\Gamma$. The description given is fairly explicit, notably in the most important special case of Hecke congruence subgroups $\Gamma$ over a polynomial ring.