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We consider solutions to the linear wave equation on a suitable globallyhyperbolic subset of an extreme Reissner-Nordstrom spacetime, arising fromregular initial data prescribed on a Cauchy hypersurface crossing the futureevent horizon. We obtain boundedness, decay, non-decay and blow-up results. Ourestimates hold up to and including the event horizon. The fundamental newaspect of this problem is the degeneracy of the redshift on the event horizon.Several new analytical features of degenerate horizons are also presented.

The Wave Equation on Extreme Reissner-Nordstr\"om Black Hole Spacetimes: Stability and Instability Results