Deligne's proof of the Weil conjecture is difficult.

On the other hand, there are some "simpler" proofs of the Weil conjecture in the case of algebraic curves.

For instance, in GTM52, one see it eventually reduced to the Hodge index theorem, which is the geometric input.

And there even exists some elementary proofs, by Bombieri or Stephanov.

So what I am asking is, will there be some "simple" proofs of the Weil conjecture for algebraic surfaces, at least for some special classes of them? And in that case, what can be the geometric inputs, without using the Standard Conjecture?