Galois properties of cohomological automorphic forms on GL\((n)\).

*(English)*Zbl 1017.11026From the introduction: Let \(F\) be a \(p\)-adic field. The author’s article [Invent. Math. 129, 75-119 (1997; Zbl 0886.11029), cited as [H1]], combined with the arguments of C. Bushnell, G. Henniart and P. Kutzko [Ann. Sci. Éc. Norm. Supér. (4) 31, 537-560 (1998; Zbl 0915.11055)], leads to a construction for a correspondence between supercuspidal representations of \(\text{GL} (n,F)\) and irreducible \(n\)-dimensional representations of the Weil group \(W_F\) of \(F\) on the \(\ell\)-adic cohomology of a certain family of rigid analytic spaces, introduced by Drinfel’d. The construction was global and as such included a determination of the local Galois representations attached to automorphic forms on certain kinds of unitary groups. These unitary groups \(GG\) are associated to division algebras with involution over CM fields \({\mathcal K}\) containing imaginary quadratic fields \({\mathcal K}_0\). Under the hypotheses of the author’s article (loc. cit.), \(p\) splits in \({\mathcal K}_0\) and the local field \(F\) occurs as the completion of \({\mathcal K}\) at a \(p\)-adic place \(v\). The local representation of \(\text{Gal} (\overline{F}/F)\) occurs in a compatible system of \(\lambda\)-adic representations of \(\text{Gal} (\overline{\mathcal K}/{\mathcal K})\), defined on the \(\ell\)-adic cohomology of a Shimura variety \(\overline{S}\) attached to \(GG\). Thus the author’s construction (loc. cit.) determines the action of the decomposition group at \(v\) on the part of the cohomology of \(\overline{S}\) corresponding to representations that are supercuspidal at \(v\).

The main purpose of the present article is to extend this result to \(\ell\)-adic cohomology of \(\overline{S}\) with coefficients in \(\ell\)-adic local systems defined by finite-dimensional algebraic representations of \(GG\). Theorem 1.7 of the author’s [Invent. Math. 134, 177-210 (1998; Zbl 0921.11060), cited as [H2]] asserts that the author’s correspondence of [H1] extends to this more general setting. This assertion is used there in [H2] to prove that the correspondence [H1] preserves \(\varepsilon\)-factors of pairs, provided the global and local correspondences on the cohomology of \(\overline{S}\) are compatible at non-supercuspidal places as well.

The proof of Theorem 1.7 of [H2] is given in §5 of the present paper. An independent proof of this theorem is given in M. Harris and R. Taylor [The geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton Univ. Press (2001; Zbl 1036.11027)]. The correspondence of [H1] is based on a Hochschild-Serre spectral sequence (Lemma 6) for the \(\ell\)-adic cohomology of \(p\)-adically uniformized Shimura varieties. This was proved using unpublished results of Berkovich, who studied the action of \(G\) on the cohomology of analytic spaces \(X\) with \(G\)-action, where \(G\) is a \(p\)-adic analytic group. Given a torsion sheaf \(F\) on \(X\) with smooth \(G\)-action, with orders prime to \(p\), the unpublished notes [V. G. Berkovich, Étale equivariant sheaves on \(p\)-adic analytic spaces, manuscript in preparation; and private communications April 1995] construct acyclic resolutions of \(F\) with smooth \(G\)-action. We apply Berkovich’s construction and show in §2 that the resolutions of the constant sheaves \(\mathbb{Z}/ \ell^m\mathbb{Z}\) satisfy a Mittag-Leffler condition as \(m\) varies. This allows us to extend Lemma 6 of [H1] to torsion sheaves and twisted \(\ell\)-adic coefficients. The construction for torsion sheaves becomes somewhat more complicated when \(\ell\) is not prime to the orders of finite subquotients of \(\text{GL} (n,F)\) (the “non-banal” case, in the terminology of M.-F. Vignéras [Représentations \(\ell\)-modulaires d’un groupe réductif \(p\)-adique avec \(\ell\neq p\), Birkhäuser, Boston (1996; Zbl 0859.22001)]).

The final section studies the \(\ell\)-torsion in the cohomology of the Drinfel’d rigid spaces when \(\ell\) is banal. Some of the results of the present paper were announced at the 1996 Durham conference on arithmetic and Galois representations. This article belongs to the author’s series aiming at a proof of the local Langlands conjecture. This has now been achieved by the author and R. Taylor and also by G. Henniart [see the overview by J. Rogawski, Notices Am. Math. Soc. 47, 35-41 (2000; Zbl 1009.11063)].

The main purpose of the present article is to extend this result to \(\ell\)-adic cohomology of \(\overline{S}\) with coefficients in \(\ell\)-adic local systems defined by finite-dimensional algebraic representations of \(GG\). Theorem 1.7 of the author’s [Invent. Math. 134, 177-210 (1998; Zbl 0921.11060), cited as [H2]] asserts that the author’s correspondence of [H1] extends to this more general setting. This assertion is used there in [H2] to prove that the correspondence [H1] preserves \(\varepsilon\)-factors of pairs, provided the global and local correspondences on the cohomology of \(\overline{S}\) are compatible at non-supercuspidal places as well.

The proof of Theorem 1.7 of [H2] is given in §5 of the present paper. An independent proof of this theorem is given in M. Harris and R. Taylor [The geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton Univ. Press (2001; Zbl 1036.11027)]. The correspondence of [H1] is based on a Hochschild-Serre spectral sequence (Lemma 6) for the \(\ell\)-adic cohomology of \(p\)-adically uniformized Shimura varieties. This was proved using unpublished results of Berkovich, who studied the action of \(G\) on the cohomology of analytic spaces \(X\) with \(G\)-action, where \(G\) is a \(p\)-adic analytic group. Given a torsion sheaf \(F\) on \(X\) with smooth \(G\)-action, with orders prime to \(p\), the unpublished notes [V. G. Berkovich, Étale equivariant sheaves on \(p\)-adic analytic spaces, manuscript in preparation; and private communications April 1995] construct acyclic resolutions of \(F\) with smooth \(G\)-action. We apply Berkovich’s construction and show in §2 that the resolutions of the constant sheaves \(\mathbb{Z}/ \ell^m\mathbb{Z}\) satisfy a Mittag-Leffler condition as \(m\) varies. This allows us to extend Lemma 6 of [H1] to torsion sheaves and twisted \(\ell\)-adic coefficients. The construction for torsion sheaves becomes somewhat more complicated when \(\ell\) is not prime to the orders of finite subquotients of \(\text{GL} (n,F)\) (the “non-banal” case, in the terminology of M.-F. Vignéras [Représentations \(\ell\)-modulaires d’un groupe réductif \(p\)-adique avec \(\ell\neq p\), Birkhäuser, Boston (1996; Zbl 0859.22001)]).

The final section studies the \(\ell\)-torsion in the cohomology of the Drinfel’d rigid spaces when \(\ell\) is banal. Some of the results of the present paper were announced at the 1996 Durham conference on arithmetic and Galois representations. This article belongs to the author’s series aiming at a proof of the local Langlands conjecture. This has now been achieved by the author and R. Taylor and also by G. Henniart [see the overview by J. Rogawski, Notices Am. Math. Soc. 47, 35-41 (2000; Zbl 1009.11063)].

##### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

22E50 | Representations of Lie and linear algebraic groups over local fields |