Let \(X\) be a curve over the finite field \(\mathbb{F}_ q\) \((q = p^ m)\) with \(\infty \in X\) a fixed closed point; let \(A\) be the affine ring of \(X - \infty\). In his fundamental paper “Elliptic modules” [Math. USSR, Sb. 23(1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014), V. G. Drinfel’d introduced elegant analogs (now called “Drinfeld modules”) of elliptic curves and abelian varieties for \(A\). Although various instances of this theory go back to L. Carlitz in the 1930’s, Drinfeld’s seminal paper marked the beginning of the modern theory of function fields over finite fields. In particular, Drinfeld was able to give a moduli theoretic construction of the maximal abelian extension of \(k\) (= fraction field of \(A)\) which is split totally at \(\infty\), as well as to give a Jacquet-Langlands style two-dimensional reciprocity law where the infinite component is a Steinberg representation. The two-dimensional reciprocity law is established by decomposing the étale cohomology of the compactified rank 2 moduli scheme. The impediment to implementing this procedure for arbitrary \(d>2\) is the difficulty of obtaining a good compactification in general.
The operator \(\tau : x \mapsto x^ q\) satisfies many analogies with the classical differentiation operator \(D : = {d \over dx}\) and these analogies go surprisingly deep. A prime example of this was given in 1976 when Drinfeld found an interpretation of a Drinfeld module \(\varphi\) in terms of a locally free sheaf \({\mathcal F}\) on \(X\) (of the same rank as \(\varphi)\) with maps relating \({\mathcal F}\) and its twist by the Frobenius map; this sheaf-theoretic interpretation being analogous to results of I. Krichever on differential operators. An excellent reference for all this is a paper by D. Mumford [in Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 115-153 (1977; Zbl 0423.14007)]. Such locally free sheaves are examples of “shtuka,” “\(F\)-sheaves,” or “elliptic sheaves” – this last being the notation used in the paper being reviewed. More generally, shtuka appear when the axioms in the “elliptic modules \(\leftrightarrow\) elliptic sheaves” dictionary are weakened a bit.
The paper under review is a very nice summary of the important work of G. Laumon, M. Rapoport and U. Stuhler on “\({\mathcal D}\)- elliptic sheaves and the Langlands correspondence” [cf. Invent. Math. 113, No. 2, 217-338 (1993)] where \({\mathcal D}\) is a central simple algebra over \(k\). These can be viewed, at least to first order, as “elliptic sheaves with complex multipliction by \({\mathcal D}\)” (and, among other axioms the dimension of \({\mathcal D}\) must also be the rank of the elliptic sheaf). A notion of “level structure” can be given generalizing that for Drinfeld modules. The point is that such \({\mathcal D}\)-elliptic sheaves have good, smooth, moduli spaces, and, when \({\mathcal D}\) is a division algebra, this moduli is projective (think of the theory of Shimura curves). Thus and importantly, the problems of noncompact moduli spaces are avoided. In particular, using the cohomology of such moduli spaces, G. Laumon, M. Rapoport and U. Stuhler prove a reciprocity law generalizing the one given in Drinfeld’s original paper. As a consequence, they find enough representations to establish the basic local Langlands conjecture for \(\text{GL}_ d\) \((d\) arbitrary) of a local field of equal characteristic; this completes a program that was first begun by P. Deligne in the 1970’s using Drinfeld’s original work for \(d = 2\).
For the entire collection see [Zbl 0772.00016].