##
**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.

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.

Reviewer: Igor Potemine (Toulouse)

### 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 |