Number Fields and Function Fields—Two Parallel Worlds

Volume 239 of the series Progress in Mathematics pp 167-222

Uniformizing the Stacks of Abelian Sheaves

  • Urs HartlAffiliated withInstitute of Mathematics, University of Freiburg

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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.