  For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$
of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology
groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic
Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is
split at $\mathfrak{p}$ and a $p$-adically admissible representation of
$\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic
deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic
representation $\pi$ of $D^*(\mathbb{A})$ this yields a natural construction of
infinitesimal deformations of the Galois representation attached to $\pi$.
