TITLE: Weil conjecture for algebraic surfaces
QUESTION [17 upvotes]: 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?

REPLY [2 votes]: I proved it for Zariski Surfaces in 1970. See chapter 1 of my book with Jeff Lang. Zariski Lives!!! 

Blass, Piotr; Lang, Jeffrey (1987), Zariski surfaces and differential equations in characteristic p>0, Monographs and Textbooks in Pure and Applied Mathematics 106, New York: Marcel Dekker Inc., ISBN 978-0-8247-7637-4, MR 879599