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Uniformizing the Stacks of Abelian Sheaves

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Book cover Number Fields and Function Fields—Two Parallel Worlds

Part of the book series: Progress in Mathematics ((PM,volume 239))

Abstract

Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon-Rapoport-Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects “of dimension 1.” Their higher dimensional generalizations are called abelian sheaves. In the analogy between function fields and number fields, abelian sheaves are counterparts of abelian varieties. In this article we study the moduli spaces of abelian sheaves and prove that they are algebraic stacks.We further transfer results of Čerednik-Drinfeld and Rapoport-Zink on the uniformization of Shimura varieties to the setting of abelian sheaves. Actually the analogy of the Čerednik-Drinfeld uniformization is nothing but the uniformization of the moduli schemes of Drinfeld modules by the Drinfeld upper half space. Our results generalize this uniformization. The proof closely follows the ideas of Rapoport-Zink. In particular, analogues of p-divisible groups play an important role. As a crucial intermediate step we prove that in a family of abelian sheaves with good reduction at infinity, the set of points where the abelian sheaf is uniformizable in the sense of Anderson, is formally closed.

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Authors and Affiliations

  1. Institute of Mathematics, University of Freiburg, Eckerstraße 1, D-79104, Freiburg, Germany

    Urs Hartl

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  1. Urs Hartl
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Editors and Affiliations

  1. Korteweg-de Vries Instituut, Universiteit van Amsterdam, 1018 TV, Amsterdam, The Netherlands

    Gerard van der Geer  & Ben Moonen  & 

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, I-00133, Roma, Italy

    René Schoof

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© 2005 Birkhäuser Boston

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Hartl, U. (2005). Uniformizing the Stacks of Abelian Sheaves. In: van der Geer, G., Moonen, B., Schoof, R. (eds) Number Fields and Function Fields—Two Parallel Worlds. Progress in Mathematics, vol 239. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4447-4_9

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