?WHAT IS... a Shtuka?

David Goss

Shtuka is a Russian word colloquially meaning thought, motivated much early work of O. Ore, “thing”. Spelled chtouca in the French literature, a E. H. Moore, and others. mathematical shtuka is, roughly speaking, a special Drinfeld Modules kind of module with a Frobenius-linear endomor- To define a Drinfeld module we need an algebra A phism (as explained below) attached to a curve over which will play the same role in the characteristic a finite field. Shtukas came from a fundamental p theory as the integers Z play in classical arith- analogy between differentiation and the p-th power metic. For simplicity of exposition we now set mapping in prime characteristic p. We will follow A = Fp[T ], the ring of polynomials in one indeter- both history and analogy in our brief presentation minate T. Let L be as above. A Drinfeld A-module here, with the hope that the reader will come to some ψ of rank d over L [Dr1] is an Fp-algebra injection appreciation of the amazing richness and beauty of ψ: A → L{τ} such that the image of a ∈ A, denoted characteristic p algebra. ψa(τ), is a polynomial in τ of degree d times the degree of a with d>0 . Note that ψ is uniquely de- Additive Polynomials termined by ψ (τ) and therefore d is a positive in- Let L be a field in characteristic p (so L is some ex- T teger. Moreover, there is a homomorphism ı from tension field of the finite field F = Z/(p)). The p A to L defined by setting ı(a) equal to the constant binomial theorem implies that the p-th power map- p term of the polynomial ψa(τ). Drinfeld modules are ping τ(x):=x satisfies τ(α + β)=τ(α)+τ(β) for α similar to elliptic curves in that they possess divi- and β in L (the coefficients of the mixed terms are sion points (= zeroes of ψ (τ)(x) for a ∈ A), Tate j pj pj j j a 0 in L); thus τ (α + β)=α + β = τ (α)+τ (β) modules, and cohomology. Moreover, like elliptic ≥ → j for any j 0. We view the mappings x τ (x) as curves, Drinfeld modules arise analytically (i.e., operators on L and on its field extensions. A poly- over the complete field Fp((1/T ))) from “lattices” m j nomial in τ is an expression p(τ):= j=0 cj τ with via an exponential function (which is an entire F - j p {c }⊆L ; so p(τ)(x)= m c xp . Like τ and τj for ∞ j j j=0 j linear function e(τ)= j=0 bj τ ). ≥ → j 0, the function x p(τ)(x) is an additive map. A Bit of Algebraic Geometry Thus its kernel, the roots of p(τ)(x) in a fixed al- For simplicity again, we now assume that L is an gebraic closure L¯ of L, is a finite-dimensional Fp- algebraically closed field. Consider the projective subspace of L¯. The set of polynomials in τ, denoted line P1 over L. An affine open subspace U of P1 is L{τ}, is a left L-vector space and forms a ring P1 minus a finite nonempty collection of points. under composition; notice that τ · (cτ)=cpτ2 , so There is a large ring Γ (U) of rational functions with this ring is not commutative in general. The anal- no poles in U. A locally free sheaf of rank d on P1 ogy with the ring of complex differential operators over L is an assignment of a free Γ (U)-module of in one variable, which becomes clear with a little rank d to each affine open subspace U in a way which is consistent with respect to the restriction David Goss is professor of mathematics at The Ohio of one affine open subspace to another. Notice State University. His email address is goss@math. that the rational functions with no poles anywhere ohio-state.edu. on P1 are the elements of L, and there are far too

36 NOTICES OF THE AMS VOLUME 50, NUMBER 1 few of these to classify locally free sheaves. There While we have worked here with A = Fp[T ], in is, however, a very clever dictionary between the lo- fact all of the above goes through readily when A cally free sheaves and certain graded modules which is replaced by the affine algebra of an arbitrary arise from homogeneous coordinates (see, e.g., §II.5 smooth projective curve X over a finite field minus of R. Hartshorne’s book Algebraic Geometry). a fixed closed point. All of the salient issues are Shtukas touched on in the simple case sketched here. The In his study of the Korteweg de Vries equation, I. M. collection of those algebraic functions on X with Krichever found a remarkable dictionary between poles of finite order forms a field k called the “func- certain sheaves on curves and subalgebras of tion field of X”. Such function fields are the analogs C[[t]][d/dt] (see, e.g., [M1]). The analogy between in finite characteristic of “number fields” defined Q τ and d/dt inspired V. G. Drinfeld to look for a sim- by adjoining to the rational numbers a finite ilar construction involving Drinfeld modules; the number of roots of polynomials with rational co- resulting sheaves will give us the shtuka. Let our efficients. Modern number theory is concerned with field L now be equipped with a Drinfeld module ψ the properties of both types of fields. of degree d. We make M := L{τ} into a module The general notion of a shtuka, which has been crucial to the work of Drinfeld and L. Lafforgue on over L ⊗F Fp[T ] L[T ] as follows: Let f (τ) ∈ M, p the Langlands conjectures for k (see [L1] and its ref- l ∈ L, and a ∈ A = Fp[T ]; we then put erences), is just the abstraction of (1) to families l ⊗ a · f (τ):=lf(ψa(τ)) U × X where U is a scheme in characteristic p (see, e.g., [L1]). Moreover, it is possible to describe which (so that elements of Fp[T ] always act via the shtukas arise from Drinfeld modules (see, e.g., ψ-action). Using a right division algorithm, one shows [M1]). readily that M is a free L[T ]-module of rank d. How- d τ-Sheaves ever, M is much richer than L[T ] because M also Over the affine line inside P1 over L, both M and { } has the left action of τ via multiplication in L τ . M reduce to M itself. The L[T,τ]-module M is · p · This action is Frobenius-linear, as τ(l m)=l τ(m) called by G. Anderson the “motive of ψ” in analogy ∈ ∈ for l L and m M. with the classical theory of motives, and its ab- The module M possesses a gradation given by the straction to families is called “τ-sheaves”. It turns degree (in τ) of an element f (τ). The action of L[T ] out that τ-sheaves are the correct notion with which given above clearly preserves this gradation. Define to describe characteristic-p-valued L-functions { ∈ | ≤ } M ⊕∞ Mj := f (τ) M degτ f (τ) j , := j=0Mj , (D. Wan–Y. Taguchi, G. Böckle–R. Pink, F. Gardeyn, M ⊕∞ M M and [1] := j=0Mj+1. Both and [1] are graded G. Böckle) and to study special values of character- modules over the graded ring constructed from L[T ] istic-p-valued Γ-functions (G. Anderson– in the same fashion as M, and they fit into the dic- W. D. Brownawell–M. Papanikolas). Moreover, τ- tionary mentioned in the preceding section. Thus sheaves are naturally associated to characteristic- both M and M[1] give rise to locally free sheaves of p-valued cusp forms (G. Böckle), much as one asso- 1  rank d on P over L, which we denote by M and M ciates elliptic curves (and other classical motives) to respectively. The mapping which injects Mj into elliptic cusp forms. Shtukas, and τ-sheaves, are such  Mj+1 gives an injection λ of M into M . Moreover, fundamental ideas that the process of mining their  multiplication by τ gives an injection of M into M riches is really just beginning! which is Frobenius linear over each affine open sub- space. We encapsulate all this by References [Dr1] V. G. DRINFELD, Elliptic modules, Math. Sbornik 94 λ  τ (1) M → M ← M . (1974), 594–627; English transl., Math. USSR Sbornik 23 (1976), 561–92. Diagram (1) is the “shtuka associated to ψ”. [L1] G. LAUMON, La correspondence de Langlands sur les The cokernel of λ gives rise to trivial modules on corps de fonctions (d’après Laurent Lafforgue), Sém. affine open subspaces not containing the point Bourbaki 873 (1999–2000). 1 ∞∈P , and the cokernel of τ also gives rise to [M1] D. MUMFORD, An algebro-geometric construction of trivial modules on affine open subspaces not con- commuting operators and solutions to the Toda taining a point lying over the prime ker ı of A. lattice equation, KdV equation and related nonlin- These are naturally called the “pole” and the “zero” ear equations, International Symposium on Algebraic of the shtuka. Geometry (Kyoto, 1977) (M. Nagata, ed.), Kinokuniya, Tokyo, 1978, pp. 115–53. When d =1, the locally free sheaves are called [Th1] D. THAKUR, Shtukas and Jacobi sums, Invent. Math. “line-bundles”, and they come from divisors. Using 111 (1993), 557–70. the Riemann-Roch Theorem and a result of Drin- feld, one can show that the shtuka actually arises from a function on P1 over L [Th1]. For instance, Comments and suggestions may be sent to the function associated to the rank 1 Drinfeld mod- [email protected]. ule C given by CT (τ):=τ is just T itself!

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