This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the [standard conjectures](http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles) on vanishing of cycles would imply the Weil conjectures. So, are there proofs of Weil conjectures in special cases using partial results on the standard conjectures? If so, which cases, and what are the references?

Background: Borcherds mentions [here](https://mathoverflow.net/questions/31963/weil-conjecture-for-algebraic-surfaces/32005#32005) that Manin proved a few special cases in higher dimensions using motives.