$$D$$ -elliptic sheaves and the langlands correspondence
$$D$$ -elliptic sheaves and the langlands correspondence
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DOI：
10.1007/bf01244308
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发表时间：
1993-12
影响因子：
3.1
通讯作者：
G. Laumon;Michael Rapoport;U. Stuhler
中科院分区：
文献类型：
--
作者：
G. Laumon;Michael Rapoport;U. Stuhler
In a series of papers [Dr 1, Dr 2], Drinfeld introduced analogues of Shimura varieties for GLe over a function field F of characteristic p> 0. Decomposing their,(-adic cohomology under the action of the Hecke operators he constructed very interesting Galois representations of F. In fact, for d= 2 he showed that the correspondence which to an automorphic representation associates the Galois representation on its eigenspace is, up to a Tate twist, a Langlands correspondence (equality of L-functions, z-factors etc.). This is completely analogous to the classical case of modular curves over Q (the Shimura variety associated to GL2). The essential difficulty in extending this result to general d lies in the non-compactness of Drinfeld's varieties. Our purpose in the present paper is to construct compact versions of Drinfeld varieties for central division algebras over F, to study their•-adic cohomology and give applications to the global and local Langlands correspondence.Drinfeld constructed his varieties as moduli spaces for two equivalent but different moduli problems, elliptic modules and elliptic sheaves. The equivalence of these two concepts was proved by Drinfeld [Dr 3] and Mumford. The idea of formulating variants of these moduli problems for a division algebra was proposed several years ago by one of us (U. St.). Here we will concentrate on the generalization of elliptic sheaves, to be called Y-elliptic sheaves. The concept of Y-elliptic module is closely related to Anderson's t-motives [An]; in spite of their elementary nature they will not play a role in this paper (comp., however, Sect. 3) since@-elliptic sheaves are easier to study.
