Maybe someone can help me out. I am considering the p-adic upper half-plane $\mathcal{H}_p$ given on points by $\mathbb{P}^1(\mathbb{C}_p)\setminus \mathbb{P}^1(\mathbb{Q}_p)$, viewed as a rigid analytic space, and I denote with $M$ the space of rigid meromorphic functions on $\mathcal{H}_p$. Then I view the multiplicative group $M^*$ as a $SL_2(\mathbb{Z})$-module by the action of $SL_2(\mathbb{Z})$ on $\mathcal{H}_p$ by Möbius-transformations, i.e. we just translate the argument. Then I try to dertermine the group $(M^*)_U$ of coinvariants of $M^*$ for a principal congruence subgroup $U$ (where we choose the $N$ big enough s.t. $U$ is torsion-free). To be more precise, I want to calculate the kernel of the projection map $\mathbb{C}_p^* \rightarrow (M^*)_U$. Has anyone an idea? Is this maybe injective? Any help would be appreciated!
Thanks for any comment :)