A proof of Ax-Grothendieck utilizes model theory and the fact that the theorem is true for finite fields, and also algebraic closures of finite fields. See here.
I have a (perhaps naive) question: considering that the Riemann hypothesis has been proved over finite fields, is it possible to utilize the approach to prove the Riemann hypothesis over $\mathbb{C}$?
I'm guessing the answer is a sound "no," otherwise it would've been done already. Where does the argument fail? Is it in writing the Riemann hypothesis in first order logic, or something with the ideas?