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

Kudla, Stephen; Rapoport, Michael

doi:10.5802/aif.2878

An alternative description of the Drinfeld

p

-adic half-plane
[Une interprétation alternative du demi-plan de Drinfeld

p

-adique]

Kudla, Stephen ¹ ; Rapoport, Michael ²

¹ University of Toronto Department of Mathematics 40 St. George St., BA6290 Toronto, Ontario, M5S 2E4 (Canada)
² Mathematisches Institut der Universität Bonn Endenicher Allee 60 53115 Bonn (Allemagne)

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1203-1228.

Résumé
Abstract

On montre que le modèle formel dû à Deligne du demi-plan $p$ -adique de Drinfeld relatif à un corps $p$ -adique $F$ représente un problème de modules de $O_{F}$ -modules munis d’une action de l’anneau des entiers dans une extension quadratique $E$ de $F$ . La démonstration repose sur une comparaison entre ce problème de modules et celui de Drinfeld des $O_{D}$ -modules formels spéciaux. Cet isomorphisme est une manifestation de l’isomorphisme exceptionel entre ${SL}_{2} (F)$ et $SU (C) (F)$ , où $C$ est un espace hermitien déployé de dimension $2$ sur $E$ .

We show that the Deligne formal model of the Drinfeld $p$ -adic half-plane relative to a local field $F$ represents a moduli problem of polarized $O_{F}$ -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 ${SL}_{2} (F)$ and $SU (C) (F)$ for a two-dimensional split hermitian space $C$ for $E / F$ .

MR Zbl

DOI : 10.5802/aif.2878

Classification : 11G18, 14G35, 11G15
Keywords: Drinfeld $p$-adic half-plane, Bruhat-Tits tree
Mot clés : demi-plan de Drinfeld $p$-adique, arbre de Bruhat-Tits

Affiliations des auteurs :

Kudla, Stephen ¹ ; Rapoport, Michael ²

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     title = {An alternative description of the {Drinfeld} $p$-adic half-plane},
     journal = {Annales de l'Institut Fourier},
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Kudla, Stephen; Rapoport, Michael. An alternative description of the Drinfeld $p$-adic half-plane. Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1203-1228. doi : 10.5802/aif.2878. http://www.numdam.org/articles/10.5802/aif.2878/

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