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

Kudla, Stephen; Rapoport, Michael

An alternative description of the Drinfeld

p

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

p

-adique]

Kudla, Stephen ; Rapoport, Michael

Annales de l'Institut Fourier, Tome 64 (2014), p. 1203-1228 / Harvested from Numdam

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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 $E / F$ .

Publié le : 2014-01-01

MR 3330168 | Zbl 06387305

DOI : https://doi.org/10.5802/aif.2878
Classification: 11G18, 14G35, 11G15
Mots clés: demi-plan de Drinfeld

p

-adique, arbre de Bruhat-Tits

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     author = {Kudla, Stephen and Rapoport, Michael},
     title = {An alternative description of the Drinfeld $p$-adic half-plane},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {1203-1228},
     doi = {10.5802/aif.2878},
     zbl = {06387305},
     mrnumber = {3330168},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_3_1203_0}
}

Kudla, Stephen; Rapoport, Michael. An alternative description of the Drinfeld $p$-adic half-plane. Annales de l'Institut Fourier, Tome 64 (2014) pp. 1203-1228. doi : 10.5802/aif.2878. http://gdmltest.u-ga.fr/item/AIF_2014__64_3_1203_0/

Bibliographie

[1] Boutot, J.-F.; Carayol, H. Uniformisation $p$ -adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfelʼd, Astérisque, Tome 7 (1991) no. 196-197, p. 45-158 (1992) (Courbes modulaires et courbes de Shimura (Orsay, 1987/1988)) | MR 1141456 | Zbl 0781.14010

[2] DrinfelʼD, V. G. Coverings of $p$ -adic symmetric domains, Funkcional. Anal. i Priložen., Tome 10 (1976) no. 2, pp. 29-40 | MR 422290 | Zbl 0346.14010

[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., Tome 184 (2011) no. 3, pp. 629-682 | Article | MR 2800697 | Zbl 1229.14020

[6] Pappas, Georgios On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom., Tome 9 (2000) no. 3, pp. 577-605 | MR 1752014 | Zbl 0978.14023

[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, International Press (Adv. Lect. in Math.) Tome 26 (2013), pp. 135-217 | MR 3135437

[8] Rapoport, Michael; Zink, Th. Period spaces for $p$ -divisible groups, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 141 (1996), pp. xxii+324 | MR 1393439 | Zbl 0873.14039

[9] Raynaud, Michel Schémas en groupes de type $(p, \dots, p)$ , Bull. Soc. Math. France, Tome 102 (1974), pp. 241-280 | Numdam | MR 419467 | Zbl 0325.14020

[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 $GU (1, s)$ , Canad. J. Math., Tome 62 (2010) no. 3, pp. 668-720 | Article | MR 2666394 | Zbl 1205.14028

[12] Vollaard, Inken; Wedhorn, Torsten The supersingular locus of the Shimura variety of $GU (1, n - 1)$ II, Invent. Math., Tome 184 (2011) no. 3, pp. 591-627 | Article | MR 2800696 | Zbl 1227.14027

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