On the Cuspidal Divisor Class Group of a Drinfeld Modular Curve

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 mo\-dular 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.

1991 Mathematics Subject Classification: 11G09, 11G18, 11F11, 11F12

Keywords: Drinfeld modular curves, theta functions, cuspidal divisor class groups

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