## On a Tower of Good Affinoids in $\mathbf{X_0(p^n)}$ and the Inertia Action on the Reduction

J. Math. Sci. Univ. Tokyo
Vol. 23 (2016), No. 1, Page 289–347.

Tsushima, Takahiro
On a Tower of Good Affinoids in $\mathbf{X_0(p^n)}$ and the Inertia Action on the Reduction
Coleman and McMurdy calculate the stable reduction of $X_0(p^3)$ for any prime number $p \geq 13$, on the basis of rigid geometry in \cite{CM}. Further, in \cite{CM2}, they compute also the inertia action on the stable reduction of $X_0(p^3)$. In \cite{T}, we have determined the stable model of $X_0(p^4)$ for any prime $p \geq 13$. In this paper, we calculate the reductions of some good'' affinoids in $X_0(p^n)$ and determine the inertia action on them. As a result, we study the middle cohomology of the reductions in terms of the type theory for $\textit{GL}_2(\mathbb{Q}_p)$ given in \cite{BHbook}.