  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.
