Survey of Drinfel’d modules. (English) Zbl 0627.14026

Deligne, Pierre; Husemoller, Dale H.

Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 25-91 (1987).

[For the entire collection see Zbl 0615.00004.]
Deligne has related two-dimensional \(\ell\)-adic representations of the Galois group \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) and “new” holomorphic modular forms of weight at least two on the upper-half plane, or equivalently, automorphic representations of the adele group \(\text{GL}_2(A_{\mathbb Q})\). The correspondence preserves \(L\)-functions. This theory depends on the properties of moduli varieties of elliptic curves with given level structure. The \(\ell\)-adic representations are realized in the \(\ell\)-adic \(H^1\) of certain sheaves. Drinfel’d transported the theory to the function field case by introducing the concept of elliptic module, which is called in the paper a Drinfel’d module, to replace elliptic curves. The paper gives a detailed survey of the Drinfel’d theory. In chapter 5, as an application, a local Langlands conjecture in finite characteristic is proved.

Reviewer: L.Vaserstein

MSC:

11G09	Drinfel’d modules; higher-dimensional motives, etc.
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11S37	Langlands-Weil conjectures, nonabelian class field theory
14H10	Families, moduli of curves (algebraic)
14L05	Formal groups, \(p\)-divisible groups
14H45	Special algebraic curves and curves of low genus
14K10	Algebraic moduli of abelian varieties, classification
14H52	Elliptic curves

Keywords:

modular varieties of elliptic curves; elliptic module; Drinfel’d module; local Langlands conjecture

Citations:

Zbl 0615.00004