From the point of view of analytic number theory the most important specific result which is proved using algebraic geometry is Burgess' bounds for character sums. The proof relies on Wiles bound for character sums, together with a rather complicated combinatorial argument. One could argue that as Stepanov, Schmidt, and Bombieri gave independent proofs of the required bounds, Weils bounds are not really required, but the "elementary" approach is certainly not easy either.