Let R = Z/p^n Z. The representation theory of GL_2 (R) over R-modules is well studied for n=1 (when R is a field) but there seems to be almost nothing known for n>1. In this talk, I will describe a single result valid for all n, and that hints at a larger story. Surprisingly, while this is a result about "modular representations" of finite groups, the proof uses the archimedean representation theory of Sp(4). I will also explain the motivation for considering this problem, which came from our attempt to solve a certain p-adic differential equation involving a particular residual cohomology class on a Siegel modular threefold. This is joint work with Atsushi Ichino.