On de Jong’s conjecture.

*(English)*Zbl 1123.11020A few years ago, A. J. de Jong proposed a significant conjecture on Galois representations of arithmetic fundamental groups of normal varieties over finite fields [Isr. J. Math. 121, 62–84 (2001; Zbl 1054.11032)]. A particular geometric version of de Jong’s conjecture can be formulated as follows:

Let \(X\) be a smooth projective curve over a finite field \(\mathbb F_q\), and let \(\rho:\pi_1(X)\to \text{GL}_n({\mathbf F})\) be a continuous representation of its arithmetic fundamental group \(\pi_1(X)\) over \({\mathbf F}=\mathbb F_\ell(t))\), with \(\mathbb F_\ell\) being another finite field of order \(\ell\) coprime to \(q\). Assume that the induced representation of the geometric fundamental group \(\pi_1(\overline X)\subset\pi_1(X)\) is absolutely irreducible. Then, under these conditions, the group \(\rho (\pi_1(\overline X))\) is finite.

As A. J. de Jong already pointed out, a proof of this conjecture would follow from a version of the geometric Langlands conjecture, the relevant formulation of which is given by the following statement:

If \(\rho:\pi_1(X) \to \text{GL}_n({\mathbf F})\) is a representation as in de Jong’s conjecture, then there exists a non-zero cuspidal Hecke eigenform \(f_\rho\) with eigenvalues corresponding to this representation \(\rho\).

In the paper under review, the author sketches a possible proof of this version of Langland’s conjecture, and therefore also a proof of the above-mentioned version of de Jong’s conjecture. However, as the author points out, his approach relies on two mathematical theories that have not yet completely worked out, and which actually do not exist in the published literature. One of them is the theory of étale sheaves with coefficients in \({\mathbf F}\), which is expected to be parallel to the corresponding theory of \({\mathbf F}'\)-adic sheaves for a local field \({\mathbf F}'\) of characteristic zero. Taking the existence of such a theory for granted, including the existence of a suitable abelian category of perverse sheaves, together with, the necessary sheaf-function correspondence, the author reformulates the above-mentioned geometric version of Langland’s conjecture in this sheaf-theoretic context, and proves it then in this form. Actually, to a representation \(\rho\) as above, he associates an object \(S_\rho\) in the derived category of the moduli stack of rank \(n\) bundles on the curve \(X\) such that \(S_\rho\) is cuspidal and satisfies the Hecke eigencondition with respect to the representation \(\rho\). Applying the assumed sheaf-function correspondence in this sheaf-theoretic framework, he then obtains a function \(f_\rho\) on \(\text{Bun}_n(\mathbb F_q)\), which indeed appears as a cuspidal Hecke eigenform with eigenvalues corresponding to the original representation \(\rho\).

This proof is carried out under the assumption that \(\ell>2n\), but it is also indicated that it works for any \(\ell\neq 2\). To summarize, the author gives a proof of the special versions of the conjectures formulated as above (Langlands, de Jong) in the case \(\ell\neq 2\), modulo the still incomplete theory of étale \({\mathbf F}\)-sheaves, which quite conceivably should exist in the form used in the present paper.

Let \(X\) be a smooth projective curve over a finite field \(\mathbb F_q\), and let \(\rho:\pi_1(X)\to \text{GL}_n({\mathbf F})\) be a continuous representation of its arithmetic fundamental group \(\pi_1(X)\) over \({\mathbf F}=\mathbb F_\ell(t))\), with \(\mathbb F_\ell\) being another finite field of order \(\ell\) coprime to \(q\). Assume that the induced representation of the geometric fundamental group \(\pi_1(\overline X)\subset\pi_1(X)\) is absolutely irreducible. Then, under these conditions, the group \(\rho (\pi_1(\overline X))\) is finite.

As A. J. de Jong already pointed out, a proof of this conjecture would follow from a version of the geometric Langlands conjecture, the relevant formulation of which is given by the following statement:

If \(\rho:\pi_1(X) \to \text{GL}_n({\mathbf F})\) is a representation as in de Jong’s conjecture, then there exists a non-zero cuspidal Hecke eigenform \(f_\rho\) with eigenvalues corresponding to this representation \(\rho\).

In the paper under review, the author sketches a possible proof of this version of Langland’s conjecture, and therefore also a proof of the above-mentioned version of de Jong’s conjecture. However, as the author points out, his approach relies on two mathematical theories that have not yet completely worked out, and which actually do not exist in the published literature. One of them is the theory of étale sheaves with coefficients in \({\mathbf F}\), which is expected to be parallel to the corresponding theory of \({\mathbf F}'\)-adic sheaves for a local field \({\mathbf F}'\) of characteristic zero. Taking the existence of such a theory for granted, including the existence of a suitable abelian category of perverse sheaves, together with, the necessary sheaf-function correspondence, the author reformulates the above-mentioned geometric version of Langland’s conjecture in this sheaf-theoretic context, and proves it then in this form. Actually, to a representation \(\rho\) as above, he associates an object \(S_\rho\) in the derived category of the moduli stack of rank \(n\) bundles on the curve \(X\) such that \(S_\rho\) is cuspidal and satisfies the Hecke eigencondition with respect to the representation \(\rho\). Applying the assumed sheaf-function correspondence in this sheaf-theoretic framework, he then obtains a function \(f_\rho\) on \(\text{Bun}_n(\mathbb F_q)\), which indeed appears as a cuspidal Hecke eigenform with eigenvalues corresponding to the original representation \(\rho\).

This proof is carried out under the assumption that \(\ell>2n\), but it is also indicated that it works for any \(\ell\neq 2\). To summarize, the author gives a proof of the special versions of the conjectures formulated as above (Langlands, de Jong) in the case \(\ell\neq 2\), modulo the still incomplete theory of étale \({\mathbf F}\)-sheaves, which quite conceivably should exist in the form used in the present paper.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

11G20 | Curves over finite and local fields |

14H25 | Arithmetic ground fields for curves |

14H30 | Coverings of curves, fundamental group |

11F80 | Galois representations |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

14A20 | Generalizations (algebraic spaces, stacks) |

11F25 | Hecke-Petersson operators, differential operators (one variable) |

##### Keywords:

curves over finite fields; Galois representations; fundamental groups; Langlands conjecture; special sheaves; stacks; derived categories; Hecke operators##### Citations:

Zbl 1054.11032**OpenURL**

##### References:

[1] | A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisque 100 (1982). · Zbl 0536.14011 |

[2] | A. Beilinson and V. Drinfeld, Chiral algebras, AMS Colloquium Publications 51, American Mathematical Society, Providence, RI, 2004. · Zbl 1138.17300 |

[3] | W. Borho and R. MacPherson, Small resolutions of nilpotent varieties, Astérisque 101–102 (1982), 23–74. |

[4] | A J. de Jong, A conjecture on arithmetic fundamental groups, Israel Journal of Mathematics 121 (2001), 61–64. · Zbl 1054.11032 |

[5] | E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture, Journal of the American Mathematical Society 15 (2002), 367–417. · Zbl 1071.11039 |

[6] | D. Gaitsgory, Automorphic sheaves and Eisenstein series, PhD Thesis, Tel Aviv University, 1997. · Zbl 0894.18007 |

[7] | D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Inventiones Mathematicae 144 (2001), 253–280. · Zbl 1072.14055 |

[8] | D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, math.AG/0204081. · Zbl 1129.11050 |

[9] | L. Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), Astérisque 82–83 (1981), 161–172. |

[10] | G. Laumon, Faisceaux automorphes pour GL n : la première construction de Drinfeld, alg-geom/9511004. |

[11] | I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, math.RT/0401222. · Zbl 1138.22013 |

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