  In this paper we study certain sheaves of $p$-adically complete rings of
differential operators on semistable models of the projective line over the
ring of integers in a finite extension $L$ of ${\mathbb Q}_p$. The global
sections of these sheaves can be identified with (central reductions of)
analytic distribution algebras of wide open congruence subgroups. It is shown
that the global sections functor furnishes an equivalence between the
categories of coherent module sheaves and finitely presented modules over the
distribution algebras. Using work of M. Emerton, we then describe admissible
representations of ${\rm GL}(2,L)$ in terms of sheaves on the projective limit
of these formal schemes.
