Fourier transforms and p-adic "Weil II"

Building on work of Crew, we give a rigid cohomological analogue of the main result of Deligne’s “Weil II”; this makes it possible to give a purely p-adic proof of the Weil conjectures. Ingredients include a p-adic analogue of Laumon’s application of the geometric Fourier transform in the l-adic setting, as well as recent results on p-adic differential equations, due to Andre, Christol, Crew, Kedlaya, Matsuda, Mebkhout, and Tsuzuki.