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 P¹(C)\P¹(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, P¹(C) is the Riemann sphere, so P¹(C)\P¹(R) is the upper half plane together with the lower half plane.

References

• Drinfeld, V. G. (1976), "Coverings of p-adic symmetric domains", Akademija Nauk SSSR. Funkcional'nyi Analiz i ego Priloženija, 10 (2): 29–40, ISSN 0374-1990, MR 0422290 
• Genestier, Alain (1996), "Espaces symétriques de Drinfeld", Astérisque (234): 124, ISSN 0303-1179, MR 1393015