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