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

Kudla, Stephen; Rapoport, Michael

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

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 ${\mathrm{SL}}_2\left(F\right)$ et $\mathrm{SU}\left(C\right)\left(F\right)$, 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 ${\mathrm{SL}}_2\left(F\right)$ and $\mathrm{SU}\left(C\right)\left(F\right)$ for a two-dimensional split hermitian space $C$ for $E/F$.

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

[1] Boutot, J.-F.; Carayol, H. Uniformisation $p$-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfelʼd, Astérisque, Volume 7 (1991) no. 196-197, p. 45-158 (1992) Courbes modulaires et courbes de Shimura (Orsay, 1987/1988) | MR | Zbl

[2] Drinfelʼd, V. G. Coverings of $p$-adic symmetric domains, Funkcional. Anal. i Priložen., Volume 10 (1976) no. 2, pp. 29-40 | MR | Zbl

[3] Kudla, Stephen; Rapoport, Michael New cases of $p$-adic uniformization (In preparation)

[4] Kudla, Stephen; Rapoport, Michael Special cycles on unitary Shimura varieties, III (In preparation)

[5] Kudla, Stephen; Rapoport, Michael Special cycles on unitary Shimura varieties I. Unramified local theory, Invent. Math., Volume 184 (2011) no. 3, pp. 629-682 | DOI | MR | Zbl

[6] Pappas, Georgios On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom., Volume 9 (2000) no. 3, pp. 577-605 | MR | Zbl

[7] Pappas, Georgios; Rapoport, Michael; Smithling, B.; Farkas, G.; Morrison, I. Local models of Shimura varieties, I. Geometry and combinatorics, Handbook of moduli, vol. III (Adv. Lect. in Math.), Volume 26, International Press, 2013, pp. 135-217 | MR

[8] Rapoport, Michael; Zink, Th. Period spaces for $p$-divisible groups, Annals of Mathematics Studies, 141, Princeton University Press, Princeton, NJ, 1996, pp. xxii+324 | MR | Zbl

[9] Raynaud, Michel Schémas en groupes de type $(p,\cdots ,p)$, Bull. Soc. Math. France, Volume 102 (1974), pp. 241-280 | Numdam | MR | Zbl

[10] Terstiege, U. Intersections of special cycles on the Shimura variety for GU$(1,2)$ (arXiv:1006.2106v1)

[11] Vollaard, Inken The supersingular locus of the Shimura variety for $\mathrm{GU}(1,s)$, Canad. J. Math., Volume 62 (2010) no. 3, pp. 668-720 | DOI | MR | Zbl

[12] Vollaard, Inken; Wedhorn, Torsten The supersingular locus of the Shimura variety of $\mathrm{GU}(1,n-1)$ II, Invent. Math., Volume 184 (2011) no. 3, pp. 591-627 | DOI | MR | Zbl

[13] Wilson, S. The supersingular locus of the Shimura variety for GU$(1,s)$ in the ramified case, 2011 (Master thesis, Bonn)