Eisenstein cohomology and the construction of mixed motives.
(Eisensteinkohomologie und die Konstruktion gemischter Motive.)

*(English)*Zbl 0795.11024
Lecture Notes in Mathematics. 1562. Berlin: Springer-Verlag. xx, 184 p. DM 52.00; öS 405.60; sFr. 57.50 /sc (1993).

Given a connected reductive algebraic group \(G\) defined over \(\mathbb{Q}\), we consider the spaces \(X_{K_ f}:= G(\mathbb{Q}) \mathbb{A}_ G(\mathbb{R})^ 0\setminus G(\mathbb{A})/ K_ \infty K_ f\) where \(K_ \infty\) is a maximal compact subgroup of \(G(\mathbb{R})\), \(A_ G\) the maximal \(\mathbb{Q}\)-split torus in the center of \(G\), and \(K_ f\) an open compact subgroup of \(G(\mathbb{A}_ f)\). Each of the (finitely many) connected components of this space is of the form \(\Gamma\setminus G(\mathbb{R})^ 0/ K_ \infty\) for a suitable arithmetic subgroup \(\Gamma\) of \(G\). Let \((\tau,E)\) be a finite dimensional algebraic representation of \(G(\mathbb{C})\); suppose that \(A_ G\) acts by a central character \(\chi_ E\) on \(E\). One is interested in the cohomology groups \(H^*(G,E)= {\displaystyle \varinjlim_{K_ f}} H^* (X_{K_ f},E)\), defined as the inductive limit over the directed system of cohomology groups \(H^* (X_{K_ f},E)\), \(K_ f\subset G(\mathbb{A}_ f)\). They carry a natural \(G(\mathbb{A}_ f)\)-module structure. One may recover the cohomology of \(X_{K_ f}\) by taking \(K_ f\)- invariants; this is endowed with a Hecke algebra module structure.

The cohomology groups \(H^*(G,E)\) have an interpretation in relative Lie algebra cohomology. This leads, by use of a result of Langlands, to a decomposition \[ H^*(G,E)= \bigoplus_{\{P\}\in{\mathcal P}} H^*(G,E)_{\{P\}} \] ranging over the set \({\mathcal P}\) of classes of associate parabolic \(\mathbb{Q}\)-subgroups of \(G\) where \(H_{\{P\}}^*\) corresponds to the space of functions of uniform moderate growth on \(G(\mathbb{Q}) A_ G(\mathbb{R})^ 0 \setminus G(\mathbb{A})\) which are negligible along \(Q\) for every parabolic \(\mathbb{Q}\)- subgroup \(Q\not\in \{P\}\), i.e., the constant term \(f_ Q\) with respect to \(Q\) is orthogonal to the space of cusp forms on the Levi component of \(Q\). The space \(H^*(G,E)_{\{G\}}\) corresponds to the space of cuspidal functions on \(G(\mathbb{Q}) A_ G(\mathbb{R})^ 0 \setminus G(\mathbb{A})\), and the theory of Eisenstein series (or residues of such) can be used to construct cohomology classes in the various cohomology spaces \(H^*(G,E)_{\{P\}}\), \(P\neq G\). The internal structure of the subspaces spanned by Eisenstein cohomology classes is closely related to the analytic properties of certain Euler products, i.e., automorphic \(L\)- functions, more general, to certain intertwining operators which naturally appear in the constant terms of the Eisenstein series used. Even in the case of groups of \(\mathbb{Q}\)-rank one, the understanding of this relationship in a very precise way is not complete.

However, it is fundamental to analyze the arithmetic nature of the Eisenstein cohomology spaces. This is what the book under review is about. The main focus is on the outline of a programme to construct mixed motives in the case of Shimura varieties. The cusp cohomology of Shimura varieties provides a certain supply of examples of motives. For the classical case \(\text{GL}_ 2/\mathbb{Q}\) the existence of motives has been proved by Eichler-Shimura. Their work is based on the congruence relation, and they established a link between the zeta functions of Shimura varieties and automorphic \(L\)-functions. For higher dimensional Shimura varieties, the congruence relation is not enough.

It is suggested in this programme to attach to specific Hecke algebra submodules in the Eisenstein cohomology (corresponding to some automorphic representation \(\pi\) of a Levi subgroup of a proper parabolic subgroup of \(G\)) a motive. Conjecturally, this motif will be an extension of pure motives, i.e. certain arithmetic objects whose existence is predicted by the Beilinson-Deligne conjectures.

This approach is discussed in detail in the cases of the projective symplectic group \(PGSp_ 2/\mathbb{Q}\) and the unitary group in three variables in chapter III. At this point, it is important to see that these ideas are consistent with existing conjectures in the theory of automorphic forms. Up to now there is no result asserting that the extensions as constructed are non-trivial. However, the vanishing of the automorphic \(L\)-function of \(\pi\) at a specific argument forces, combined with some additional arithmetic information, the existence of these motives.

Chapter IV contains a slightly different way of constructing mixed motives; the case dealt with are Anderson’s mixed motives. Following the Beilinson-Deligne conjecture, they correspond to the values of the Riemann zeta function at odd positive or even integer arguments. Here the Hodge-de Rham extension class of this motif (being an extension of a Tate motie) is determined and related to the appropriate value of the zeta function.

Chapter I is a well suited introduction to the Beilinson-Deligne conjectures, presented in their motivic version.

The appendix contains a letter of the author discussing the topological trace formula for Hecke operators on the cohomology of arithmetic groups as developed by M. Goresky and R. McPherson (and J. Bewersdorff in a simple case). This may be used in certain situations as a substitute for Arthur’s trace formula to avoid any analytic difficulties arising therein. The main objective here is to show that the topological trace formula sheds some light on the existence of mixed motives in Shimura varieties.

The cohomology groups \(H^*(G,E)\) have an interpretation in relative Lie algebra cohomology. This leads, by use of a result of Langlands, to a decomposition \[ H^*(G,E)= \bigoplus_{\{P\}\in{\mathcal P}} H^*(G,E)_{\{P\}} \] ranging over the set \({\mathcal P}\) of classes of associate parabolic \(\mathbb{Q}\)-subgroups of \(G\) where \(H_{\{P\}}^*\) corresponds to the space of functions of uniform moderate growth on \(G(\mathbb{Q}) A_ G(\mathbb{R})^ 0 \setminus G(\mathbb{A})\) which are negligible along \(Q\) for every parabolic \(\mathbb{Q}\)- subgroup \(Q\not\in \{P\}\), i.e., the constant term \(f_ Q\) with respect to \(Q\) is orthogonal to the space of cusp forms on the Levi component of \(Q\). The space \(H^*(G,E)_{\{G\}}\) corresponds to the space of cuspidal functions on \(G(\mathbb{Q}) A_ G(\mathbb{R})^ 0 \setminus G(\mathbb{A})\), and the theory of Eisenstein series (or residues of such) can be used to construct cohomology classes in the various cohomology spaces \(H^*(G,E)_{\{P\}}\), \(P\neq G\). The internal structure of the subspaces spanned by Eisenstein cohomology classes is closely related to the analytic properties of certain Euler products, i.e., automorphic \(L\)- functions, more general, to certain intertwining operators which naturally appear in the constant terms of the Eisenstein series used. Even in the case of groups of \(\mathbb{Q}\)-rank one, the understanding of this relationship in a very precise way is not complete.

However, it is fundamental to analyze the arithmetic nature of the Eisenstein cohomology spaces. This is what the book under review is about. The main focus is on the outline of a programme to construct mixed motives in the case of Shimura varieties. The cusp cohomology of Shimura varieties provides a certain supply of examples of motives. For the classical case \(\text{GL}_ 2/\mathbb{Q}\) the existence of motives has been proved by Eichler-Shimura. Their work is based on the congruence relation, and they established a link between the zeta functions of Shimura varieties and automorphic \(L\)-functions. For higher dimensional Shimura varieties, the congruence relation is not enough.

It is suggested in this programme to attach to specific Hecke algebra submodules in the Eisenstein cohomology (corresponding to some automorphic representation \(\pi\) of a Levi subgroup of a proper parabolic subgroup of \(G\)) a motive. Conjecturally, this motif will be an extension of pure motives, i.e. certain arithmetic objects whose existence is predicted by the Beilinson-Deligne conjectures.

This approach is discussed in detail in the cases of the projective symplectic group \(PGSp_ 2/\mathbb{Q}\) and the unitary group in three variables in chapter III. At this point, it is important to see that these ideas are consistent with existing conjectures in the theory of automorphic forms. Up to now there is no result asserting that the extensions as constructed are non-trivial. However, the vanishing of the automorphic \(L\)-function of \(\pi\) at a specific argument forces, combined with some additional arithmetic information, the existence of these motives.

Chapter IV contains a slightly different way of constructing mixed motives; the case dealt with are Anderson’s mixed motives. Following the Beilinson-Deligne conjecture, they correspond to the values of the Riemann zeta function at odd positive or even integer arguments. Here the Hodge-de Rham extension class of this motif (being an extension of a Tate motie) is determined and related to the appropriate value of the zeta function.

Chapter I is a well suited introduction to the Beilinson-Deligne conjectures, presented in their motivic version.

The appendix contains a letter of the author discussing the topological trace formula for Hecke operators on the cohomology of arithmetic groups as developed by M. Goresky and R. McPherson (and J. Bewersdorff in a simple case). This may be used in certain situations as a substitute for Arthur’s trace formula to avoid any analytic difficulties arising therein. The main objective here is to show that the topological trace formula sheds some light on the existence of mixed motives in Shimura varieties.

Reviewer: J.Schwermer (Eichstätt)

##### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F75 | Cohomology of arithmetic groups |

11F80 | Galois representations |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14G35 | Modular and Shimura varieties |

14F99 | (Co)homology theory in algebraic geometry |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |