HST/STIS Lyman-

Lyman-α (Lyα) emission of neutral hydrogen (λ1215.67 Å) is the main contributor to the ultraviolet flux of low-mass stars such as M dwarfs. It is also the main light source used in studies of the evaporating upper atmospheres of transiting extrasolar planets with ultraviolet transmission spectroscopy. However, there are very few observations of the Lyα emissions of quiet M dwarfs, and none exist for those hosting exoplanets. Here, we present Lyα observations of the hot-Neptune host star GJ 436 with the Hubble Space Telescope Imaging Spectrograph (HST/STIS). We detect bright emission in the first resolved and high quality spectrum of a quiet M dwarf at Lyα. Using an energy diagram for exoplanets and an N-body particle simulation, this detection enables the possible exospheric signature of the hot Neptune to be estimated as a ~11% absorption in the Lyα stellar emission, for a typical mass-loss rate of 1010 g s-1. The atmosphere of the planet GJ 436b is found to be stable to evaporation, and should be readily observable with HST. We also derive a correlation between X-ray and Lyα emissions for M dwarfs. This correlation will be useful for predicting the evaporation signatures of planets transiting other quiet M dwarfs