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

Stephen Kudla[1]; Michael Rapoport[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 (2014)

  • Volume: 64, Issue: 3, page 1203-1228
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Kudla, Stephen, and Rapoport, Michael. "An alternative description of the Drinfeld $p$-adic half-plane." Annales de l’institut Fourier 64.3 (2014): 1203-1228. <http://eudml.org/doc/275434>.

@article{Kudla2014,
abstract = {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 $\{\rm SL\}_2(F)$ and $\{\rm SU\}(C)(F)$ for a two-dimensional split hermitian space $C$ for $E/F$.},
affiliation = {University of Toronto Department of Mathematics 40 St. George St., BA6290 Toronto, Ontario, M5S 2E4 (Canada); Mathematisches Institut der Universität Bonn Endenicher Allee 60 53115 Bonn (Allemagne)},
author = {Kudla, Stephen, Rapoport, Michael},
journal = {Annales de l’institut Fourier},
keywords = {Drinfeld $p$-adic half-plane; Bruhat-Tits tree; Drinfeld half plane; local fields; Drinfeld moduli problem},
language = {eng},
number = {3},
pages = {1203-1228},
publisher = {Association des Annales de l’institut Fourier},
title = {An alternative description of the Drinfeld $p$-adic half-plane},
url = {http://eudml.org/doc/275434},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Kudla, Stephen
AU - Rapoport, Michael
TI - An alternative description of the Drinfeld $p$-adic half-plane
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 1203
EP - 1228
AB - 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 ${\rm SL}_2(F)$ and ${\rm SU}(C)(F)$ for a two-dimensional split hermitian space $C$ for $E/F$.
LA - eng
KW - Drinfeld $p$-adic half-plane; Bruhat-Tits tree; Drinfeld half plane; local fields; Drinfeld moduli problem
UR - http://eudml.org/doc/275434
ER -

References

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  2. V. G. Drinfelʼd, Coverings of p -adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), 29-40 Zbl0346.14010MR422290
  3. Stephen Kudla, Michael Rapoport, New cases of p -adic uniformization 
  4. Stephen Kudla, Michael Rapoport, Special cycles on unitary Shimura varieties, III 
  5. Stephen Kudla, Michael Rapoport, Special cycles on unitary Shimura varieties I. Unramified local theory, Invent. Math. 184 (2011), 629-682 Zbl1229.14020MR2800697
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  7. Georgios Pappas, Michael Rapoport, B. Smithling, Local models of Shimura varieties, I. Geometry and combinatorics, Handbook of moduli, vol. III 26 (2013), 135-217, FarkasG.G. Zbl1322.14014MR3135437
  8. Michael Rapoport, Th. Zink, Period spaces for p -divisible groups, 141 (1996), Princeton University Press, Princeton, NJ Zbl0873.14039MR1393439
  9. Michel Raynaud, Schémas en groupes de type ( p , ⋯ , p ) , Bull. Soc. Math. France 102 (1974), 241-280 Zbl0325.14020MR419467
  10. U. Terstiege, Intersections of special cycles on the Shimura variety for GU ( 1 , 2 )  Zbl06237588
  11. Inken Vollaard, The supersingular locus of the Shimura variety for GU ( 1 , s ) , Canad. J. Math. 62 (2010), 668-720 Zbl1205.14028MR2666394
  12. Inken Vollaard, Torsten Wedhorn, The supersingular locus of the Shimura variety of GU ( 1 , n - 1 ) II, Invent. Math. 184 (2011), 591-627 Zbl1227.14027MR2800696
  13. S. Wilson, The supersingular locus of the Shimura variety for GU ( 1 , s ) in the ramified case, (2011) 

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