Drinfeld upper half plane
# Drinfeld upper half plane

In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by Drinfeld (1976). It is defined to be $P^1(C)\P^1(F_∞),$ where F is a function field of a curve over a finite field, $F_∞$ its completion at ∞, and C the completion of the algebraic closure of $F_∞.$

The analogy with the usual upper half plane arises from the fact that the global function field F is analogous to the rational numbers Q. Then, $F_∞$ is the real numbers R and the algebraic closure of $F_∞$ is the complex numbers C (which are already complete). Finally, $P^1(C)$ is the Riemann sphere, so $P^1(C)\P^1(R)$ is the upper half plane together with the lower half plane.

