An alternative description of the Drinfeld $p$-adic half-plane

Abstract

We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a local field F represents a moduli problem of polarized OF -modules with an action of the ring of integers in a quadratic extension E of F . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL2(F ) and SU(C)(F ) for a two-dimensional split hermitian space C for E/F .