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Abstract

Drinfeld modular forms were introduced by D. Goss in 1980 for congruence subgroups of ${\rm GL}_2(\mathbb{F}_q[T])$. They are a counterpart of classical modular forms in the function field world. In this thesis I study Drinfeld modular forms for inner forms of ${\rm GL}_2$ that correspond to unit groups $\Lambda^\star$ of quaternion division algebras over $\mathbb{F}_q(T)$ split at the place $\infty = 1/T$. I show, following work of Teitelbaum for ${\rm GL}_2(\mathbb{F}_q[T])$, that these forms have a combinatorial interpretation as certain maps from the edges of the Bruhat-Tits tree $\mathcal{T}$ associated to ${\rm PGL}_2(K_\infty)$. Here $K_\infty$ denotes the completion of $K$ at $\infty$. A major focus of this thesis is on computational aspects: I present an algorithm for computing a fundamental domain for the action of $\Lambda^\star$ on $\mathcal{T}$ with an edge pairing, and describe how to obtain a basis of the space of these forms out of this fundamental domain. On this basis one can compute the Hecke action.