It has been assumed that Himalayan earthquakes are driven by the release of compressional strain accumulating close to the Greater Himalaya. However, elastic models of the Indo-Asian collision using recently imaged subsurface interface geometries suggest that a substantial fraction of the southernmost 500 kilometres of the Tibetan plateau participates in driving great ruptures. We show here that this Tibetan reservoir of elastic strain energy is drained in proportion to Himalayan rupture length, and that the consequent growth of slip and magnitude with rupture area, when compared to data from recent earthquakes, can be used to infer a approximately 500-year renewal time for these events. The elastic models also illuminate two puzzling features of plate boundary seismicity: how great earthquakes can re-rupture regions that have already ruptured in recent smaller earthquakes and how mega-earthquakes with greater than 20 metres slip may occur at millennia-long intervals, driven by residual strain following many centuries of smaller earthquakes.