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Drinfeld shtukas and Langlands correspondence. (Chtoucas de Drinfeld et correspondance de Langlands.) (French) Zbl 1038.11075
In this outstanding paper, the author proves the famous Langlands conjecture for $$\text{GL}_r$$ over global function fields.
Let $$X$$ be a smooth projective geometrically irreducible curve over a finite field $$\mathbb{F}_q$$ and $$F$$ its function field. Denote by $$G_F$$ the Galois group of $$F$$, $$| X|$$ the set of closed points of $$X$$, $$\mathbb{A}=\prod_{x\in| X| }F_x$$ the ring of adèles, and $$O_{\mathbb{A}}$$ the subring of integral adèles. Fix a prime number $$\ell$$ not dividing $$q$$. Firstly, for any integer $$r\geq1$$ consider the set $$\mathcal{A}^r(F)$$ of irreducible cuspidal representations $$\pi$$ of $$\text{GL}_r(\mathbb{A})$$ with central character $$\chi_{\pi}$$ of finite order. Secondly, consider the set $$\mathcal{G}_{\ell}^r(F)$$ of irreducible almost everywhere non-ramified $$\ell$$-adic representations of $$G_F$$ of dimension $$r$$ whose determinants are of finite order.
Theorem. Let $$r\geq1$$.
(i)$$_r$$ To any cuspidal automorphic representation $$\pi\in\mathcal{A}^r(F)$$ one can associate a unique Galois representation $$\sigma_{\pi}\in \mathcal{G}_{\ell}^r(F)$$ with the same ramification locus in $$| X|$$ and such that Frobenius eigenvalues of $$\sigma_{\pi}$$ coincide with Hecke eigenvalues $$z_1(\pi_x),\cdots,z_r(\pi_x)$$ for any (non-ramified) place $$x\in| X|$$.
(ii)$$_r$$ Reciprocally, to any Galois representation $$\sigma\in \mathcal{G}_{\ell}^r(F)$$, one can associate a unique automorphic representation $$\pi_{\sigma}\in\mathcal{A}^r(F)$$ whose Hecke eigenvalues coincide with Frobenius eigenvalues of $$\sigma$$.
The unicity follows from Chebotarev’s density theorem and “strong multiplicity one” theorem of Piatetski-Shapiro. In addition, for $$r=1$$ the theorem is equivalent to the reciprocity law of the class field theory.
The first key point of Lafforgue’s proof is an application of the “Deligne recursive principle”. Fix a rank $$r\geq 2$$ and suppose that assertion (i)$$_r$$ is true for all ranks $$<r$$. Then, combining functional equations for $$L$$-functions, Laumon’s product formula and reciprocity theorems of Weil, Hecke and Piatetski-Shapiro, Lafforgue (following Deligne) shows that assertion (ii)$$_r$$ is true for all ranks $$\leq r$$.
Thus, this argument reduces the proof to a construction of an appropriate map $$\mathcal{A}^r(F)\rightarrow\mathcal{G}_{\ell}^r(F)$$, $$\pi\mapsto \sigma_{\pi}$$.
This map is constructed by a careful study of the cohomology of the moduli stack $$\text{Cht}_N^r$$ of Drinfeld shtukas of rank $$r$$ with (an appropriate) level structure $$N$$. This stack is of relative dimension $$2r-2$$ and smooth over $$(X-N)\times(X-N)$$. The Hecke algebra $$\mathcal{H}_N^r$$ and two “partial Frobenius” morphisms, denoted $$\text{Frob}_{\infty}$$ and $$\text{Frob}_0$$, act on $$\text{Cht}_N^r$$. The main difficulty is that connected components of $$\text{Cht}_N^r$$ are not of finite type. Their $$\ell$$-adic cohomology are infinite-dimensional and the numbers of fixed points with respect to Hecke correspondences are also infinite.
Choose an idèle $$a\in\mathbb{A}^{\times}$$ of degree 1 and consider the quotient $$\text{Cht}_N^r/a^{\mathbb{Z}}\simeq \coprod_{1\leq d\leq r} \text{Cht}_N^{r,d}$$. Here $$\text{Cht}_N^{r,d}$$ denotes the moduli stack of shtukas of rank $$r$$ and degree $$d$$ with level structure $$N$$. Now one has a new difficulty since there is no open substack of finite type in $$\text{Cht}_N^r$$ stable by actions of $$\text{Frob}_{\infty}$$, $$\text{Frob}_0$$ and $$\mathcal{H}_N^r$$.
Nevertheless, Lafforgue had an ingenious idea to bound the canonical Harder-Narasimhan polygons $$p_{\text{HN}}^{}$$ of shtukas by a fixed polygon $$p:[0,r]\rightarrow\mathbb{R}_+$$. The corresponding stack $\text{Cht}_N^{r,p_{\text{HN}}^{}\leqslant p}/a^{\mathbb{Z}}\simeq \coprod_{1\leq d\leq r} \text{Cht}_N^{r,d,p_{\text{HN}}^{}\leqslant p}$ is of finite type, and it is possible to calculate the number of fixed points. However, $$\text{Cht}_N^{r,p_{\text{HN}}^{}\leqslant p}$$ is not stable with respect to Hecke correspondences.
In order to recover the action of Hecke correspondences it is necessary to consider a compactified stack $$\overline{\text{Cht}_N^{r,p_{\text{HN}}^{}\leqslant p}}$$. It is constructed as the moduli stack of “iterated shtukas” of rank $$r$$ with level structure $$N$$ etc. The next ingenious idea of Lafforgue is to consider iterated shtukas whose “degenerators” avoid $$N$$. Then, remarkably, normalized Hecke correspondences stabilize the corresponding open substack $$\overline{\text{Cht}_N^{r,p_{\text{HN}}^{}\leqslant p}}'$$.
Finally, in a wonderful manner, Lafforgue subtracts the same essential part $$H_{N,\text{ess}}$$ from $H_c^{2r-2} (\overline{\text{Cht}_N^{r, p_{\text{HN}}^{}\leqslant p}}'/a^{\mathbb{Z}}), \quad H_c^{2r-2}( \text{Cht}_N^{r,p_{\text{HN}}^{}\leqslant p}/a^{\mathbb{Z}})$ as well as from $$H_c^{2r-2} (\text{Cht}_N^r/a^{\mathbb{Z}})$$. The decomposition of $$H_{N,\text{ess}}$$ considered as a $$(\mathcal{H}_N^r\times G_F)$$-bimodule permits him to construct $$\sigma_{\pi}$$ and to prove the global Langlands conjecture.
Obviously, only the biggest landmarks are mentioned here, while Lafforgue’s paper contains a whole bunch of ingenious constructions, ideas and results.

##### MSC:
 11R39 Langlands-Weil conjectures, nonabelian class field theory 11F52 Modular forms associated to Drinfel’d modules 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14G35 Modular and Shimura varieties 14H60 Vector bundles on curves and their moduli 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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