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Polylogarithms and motivic Galois groups. (English) Zbl 0842.11043

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 43-96 (1994).
Let \(F\) be any field. Beilinson conjectured the existence of a mixed Tate category \({\mathcal M}_T(F)\) of mixed motivic Tate sheaves of \(\text{Spec} (F)\) relating the algebraic \(K\)-theory of \(F\) to certain extensions in this category by \[ \text{Ext}^i_{{\mathcal M}_T (F)}(\mathbb{Q} (0), \mathbb{Q}(n))= \text{gr}^n_\gamma K_{2n-i} (F)_\mathbb{Q}, \] where \(\gamma\) is the usual \(\gamma\)-filtration on \(K\)-groups. The category \({\mathcal M}_T(F)\) is a tannakian \(\mathbb{Q}\)-linear category with ‘Tate object’ \(\mathbb{Q} (1)\) satisfying suitable conditions on the Hom’s and Ext’s. Any simple object must be a tensor power of the Tate object. Furthermore, any object \({\mathcal F}\) of \({\mathcal M}_T(F)\) admits a canonical finite increasing filtration such that the \(i\)-th graded piece is isomorphic to a direct sum of \(\mathbb{Q}(-i)\)’s. The \(\text{Hom} (\mathbb{Q} (-i),-)\) give a fiber functor to graded \(\mathbb{Q}\)-vector spaces. Write \(L(F)_\bullet\) for the graded pro-Lie algebra of all derivations of the fiber functor. It is concentrated in negative degrees: \(L(F)_\bullet= \bigoplus^\infty_{n=1} L(F)_{-n}\). Its cohomology \(H^i(L (F)_\bullet)= H^i (L(F)_\bullet, \mathbb{Q})\) is graded in positive degrees, and Beilinson’s condition means that the degree \(n\) part \(H^i_{(n)} (L(F)_\bullet)\) equals \(\text{gr}^n_\gamma K_{2n-i}(F)_\mathbb{Q}\).
The whole formalism must be compatible with products of the various notions and should be functorial with respect to field embeddings. \(L(F)_\bullet\) is the Lie algebra of a pro-unipotent group \(G({\mathcal M}_T(F))\) and by definition the motivic Galois group \(\text{Gal} ({\mathcal M}_T (F))\) is the semi-direct product \(\mathbb{G}_m\times G({\mathcal M}_T (F))\), where \(\mathbb{G}_m\) accounts for the grading.
The set-up given so far leads already to interesting reformulations of e.g. the Beilinson-Soulé conjecture on the one hand and Suslin’s result on the other hand on \(\text{gr}^n_\gamma K_m (F)_\mathbb{Q}\), and with respect to Milnor \(K\)-theory implies that \(H^n_{(n)} (L(F))= K^M_n (F)_\mathbb{Q}\). Conversely, when \(F\) is a number field, results on \(K_m (F)\) imply that \(H^i (L (F)_\bullet )=0\) for \(i\geq 2\), and hence that \(L(F)_\bullet\) is a free graded Lie algebra. For a finite field \(F\), \(K_* (F)_\mathbb{Q} =0\) and thus \(L(F)_\bullet =0\), in agreement with the fact that \({\mathcal M}_T (F)\) should be semi-simple in this case.
The purpose of the paper is to state a precise conjecture on \(\text{Gal} ({\mathcal M}_T (F))\), or the associated \(L(F)_\bullet\), and to see how this conjecture is related to the construction of complexes \(\Gamma (F, n)_\mathbb{Q}\) that should satisfy all the Beilinson-Lichtenbaum axioms (modulo torsion). For a number field \(F\) this would imply Zagier’s conjecture on the value of the Dedekind zeta-function \(\zeta_F (s)\) at integer points \(n\) in terms of a determinant whose entries are rational linear combinations of values of the classical \(n\)-polylogarithms at (the complex embeddings of) some elements of this field. This was proved for \(n=2\) by Zagier. Here a proof for \(n=3\) is given.
First, define \(I(F)_\bullet:= \bigoplus^\infty_{n=2} L(F)_-\). Then, inductively, one defines subgroups \({\mathcal R}_n (F) \subset \mathbb{Z} [\mathbb{P}^1_F ]\), \(n\geq 1\), and one sets \({\mathcal B}_n (F):= \mathbb{Z} [\mathbb{P}^1_F ]/ {\mathcal R}_n (F)\). For \(n=1\) one defines \({\mathcal R}_1 (F):= (\{x\}+ \{y\}- \{xy\})\), \((x,y\in F^\times; \{0\}; \{\infty \})\), thus \({\mathcal B}_1 (F)= F^\times\). Here \(\{x\}, \dots\), are symbols. The higher \({\mathcal R}_n (F)\) are defined via homomorphisms \(\delta_n: \mathbb{Z}[\mathbb{P}^1_F ]\to {\mathcal B}_{n-1} (F) \otimes F^\times\) for \(n\geq 3\) (\(\wedge^2 F^\times\) for \(n=2\), respectively) in such a way that \(\delta_n ({\mathcal R}_n (F)) =0\); thus one gets \(\delta: {\mathcal B}_n (F)\to {\mathcal B}_{n-1} (F)\otimes F^\times\) (\(\wedge^2 F^\times\) respectively).
In terms of these objects the main conjecture can be stated: For any field \(F\) one has (i) \(I(F)_\bullet\) is a free graded pro-Lie algebra; (ii) \(H^1_{(n)} (I (F)_\bullet )\simeq {\mathcal B}_n (F)_\mathbb{Q}\), \(n\geq 2\), i.e. \(I(F)_\bullet\) is generated as a graded Lie algebra by the spaces \({\mathcal B}_n (F)^\vee\) sitting in degree \(-n\); (iii) \(L_\bullet/ I_\bullet \simeq (F^\times_\mathbb{Q} )^\vee\) and the maps \(\delta\) coincide with maps \(f_n: H^1_{(n)} (I_\bullet)\to H^1_{(n-1)} (I_\bullet) \otimes F^\times_\mathbb{Q}\), \(n\geq 3\) \((\wedge^2 F^\times_\mathbb{Q}\), \(n=2)\). Much of the formulation of the conjecture comes from Zagier’s conjecture on polylogarithms in case \(F\) is a number field as stated by Zagier himself and later by Beilinson and Deligne in a motivic setting. Using a Hochschild-Serre spectral sequence for the ideal \(I_\bullet\) can define the complex \(\Gamma (F,n)\) as \[ \Gamma (F,n): {\mathcal B}_n @>\delta{\;}>> {\mathcal B}_{n-1} \otimes F^\times @> \delta{\;}>> {\mathcal B}_{n-2} \otimes \wedge^2 F^\times @> \delta{\;}>> \cdots @> \delta{\;}>> {\mathcal B}_2 \otimes \wedge^{n-2}F^\times @>\delta{\;}>> \wedge^n F^\times \] with \({\mathcal B}_n= {\mathcal B}_n (F)\) placed in degree 1. The main conjecture now implies that \(H^i (\Gamma (F,n)_\mathbb{Q}) \simeq \text{gr}^n_\gamma K_{2n-i} (F)_\mathbb{Q}\). In particular, for \(i=1\), one should have \(H^1(\Gamma (\mathbb{C},n)_\mathbb{Q})= \text{Ker} ({\mathcal B}_n(\mathbb{C})_\mathbb{Q} @> \delta{\;}>> {\mathcal B}_{n-1} (\mathbb{C}_\mathbb{Q} \otimes \mathbb{C}^\times)) \simeq \text{gr}^n_\gamma K_{2n-1} (\mathbb{C})_\mathbb{Q}\). Zagier’s polylogarithm \({\mathcal L}_n\) gives a homomorphism \({\mathcal L}_n: {\mathcal B}_n (\mathbb{C})\to \mathbb{R}\) and it is expected that its restriction to \(H^1 (\Gamma (\mathbb{C}, n)_\mathbb{Q}) \subset {\mathcal B}_n (\mathbb{C})_\mathbb{Q}\) coincides with the Borel regulator. Then, combining with results of Borel, Zagier’s conjecture follows: With the usual notation one has \[ \zeta_F (n)= \pi^{(r_1+ 2r_2- d_n) n} |d_F |^{-1/2} \text{ det } |{\mathcal L}_n (\sigma_j (y_i))| \] for suitable elements \(y_1, \dots, y_{d_n}\in \text{Ker} ({\mathcal B}_n (F)_\mathbb{Q} @>\delta{\;}>> {\mathcal B}_{n-1} (F)_\mathbb{Q} \otimes F^\times_\mathbb{Q})\).
Then the heart of the paper is concerned with a proof of Zagier’s conjecture for \(n=3\). It takes about twelve pages and is of a technical geometric nature. A generalized cross-ratio for a configuration of six points in \(\mathbb{P}^2\) is introduced and a seven term functional equation for the trilogarithm is derived. The strategy is to construct a homomorphism \(c_5: K_5(F)\to H^1 (\Gamma (F,3)_\mathbb{Q})\) and to show that the composition \(K_5 (\mathbb{C}) @>c_5>> H^1 (\Gamma (\mathbb{C}, 3)_\mathbb{Q} )\to \mathbb{R}\), where the last map is essentially \({\mathcal L}_3\), coincides with the Borel regulator.
The next (i.e. fourth) section is concerned with some arguments for the main conjecture. Using the generalized cross-ratio, groups \(R_2 (F)\) and \(R_3 (F)\) are introduced and, analogous to the \({\mathcal B}_n (F)\), quotients \(B_2 (F)\) and \(B_3 (F)\). The corresponding complexes \(B_F (2)\) and \(B_F(3)\) are defined also analogously to the aforementioned. Thus \(B_F (2)\) is the Bloch-Suslin complex. One has an isomorphism \(B_2 (F) \simeq {\mathcal B}_2 (F)\) by results of Suslin, and it is expected that \(B_3 (F) \simeq {\mathcal B}_3 (F)\). In fact, the proof of Zagier’s conjecture gave considerable evidence for an isomorphism \(H^i (B_F (3)_\mathbb{Q}) \simeq H^i (\wedge^\bullet_{(3)} L(F)^\vee_\bullet)\), where the subscript (3) means the degree three subcomplex of the cochain complex of the Lie algebra \(L(F)_\bullet\). On the level of complexes the situation becomes different for \(n\geq 4\), but for \(n=4\) at least there is an isomorphism in cohomology. For higher \(n\) things look even more complicated. One expects the \(\Gamma (F, N)_\mathbb{Q}\) to satisfy the Beilinson-Lichtenbaum axioms, in particular, there should be tensor products of motivic complexes \(\Gamma (n) \otimes^\mathbb{L} \Gamma (m)\to \Gamma (m+n)\) defined in the derived category. For the complexes \(\Gamma (F, n)\) a natural tensor product exists in the derived category only and not at the level of complexes. Some ideas on the construction of the groups \(L(F)_{-4}\) and \(L(F)_{-5}\) in relation to the \(B\)’s and the \({\mathcal B}\)’s lead to the belief that something similar can be done for all \(L(F)_{-n}\) and that one will be able to obtain functional equations for the higher polylogarithms and an explicit definition of the tensor product (in the derived category) of motivic complexes \(\Gamma (F, n)\).
The fifth section is concerned with explicit formulas for the universal Chern class \(c_3\) in motivic and Deligne cohomology. For any complex algebraic manifold \(X\) a regulator \(r_3: H^\bullet_{\mathcal M} (X, \mathbb{Z} (3))\to H^\bullet_{\mathcal D} (X,\mathbb{R})\) is constructed.
For the entire collection see [Zbl 0788.00054].

MSC:

11R70 \(K\)-theory of global fields
14F99 (Co)homology theory in algebraic geometry
19D99 Higher algebraic \(K\)-theory
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