Is there any analogue of Morse theory in Number theory? Naive idea arising in my head is that defining a Morse function on scheme and find etale cohomology using that function. Since I'm not an expert about algebraic geometry and Morse theory, I can't advance my thoughts.
One usually considers the analogue of Morse theory in algebraic geometry to be the theory of vanishing cycles and Lefschetz pencils.
Because of the nature of algebraic functions, Morse theory must be a little more complicated. A Morse function on a compact manifold lets us build the manifold up step by step, starting with a local minimum from which the manifold "springs from nothing". Such maps do not exist in complex geometry or algebraic geometry.
Instead, Lefschetz considers a map from a smooth projective variety to $\mathbb P^1$. One demands a Morselike condition that the critical points of this map are as simple as possible. Applying vanishing cycles theory one can relate the cohomology of the total space to the critical points but one also needs to know as a starting point the cohomology of a smooth fiber.
These were adapted to etale cohomology in SGA 72 by Deligne and Katz and were crucial in Deligne's proof of Weil's Riemann hypothesis, so this certainly gives a connection to number theory.

$\begingroup$ Thank you for clear answer. Actually, there is one more question I have. As this MO question, can we obtain similar results (such as duality, cup product, isomorphism, ...) via Lefschetz pencils? $\endgroup$ Mar 15 '17 at 5:04