  We derive the precise late-time asymptotics for solutions to the wave
equation on extremal Reissner-Nordstr\"om black holes and explicitly express
the leading-order coefficients in terms of the initial data. Our method is
based on purely physical space techniques. We derive novel weighted energy
hierarchies and develop a singular time inversion theory, which allow us to
uncover the subtle contribution of both the near-horizon and near-infinity
regions to the precise asymptotics. We introduce a new horizon charge and
provide applications pertaining to the interior dynamics of extremal black
holes. Our work confirms, and in some cases extends, the numerical and
heuristic analysis of Lucietti-Murata-Reall-Tanahashi, Ori-Sela and
Blaksley-Burko.
