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Motivic interpretation of the Zagier conjecture relating polylogarithms and regulators. (Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs.) (French) Zbl 0799.19004
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, 97-121 (1994).
For a smooth \(\mathbb{Q}\)-scheme \(S\) the Beilinson conjectures are concerned with the study of the regulator map \[ \text{reg}:\bigl( K_{2k-1} (S) \otimes \mathbb{Q} \bigr)^{(k)} \to \text{Ext}^ 1_{S(\mathbb{C})} \bigl( \mathbb{Q} (0), \mathbb{Q} (k) \bigr), \] where \(K_ i (S) \otimes \mathbb{Q}^{(j)}\) (called motivic cohomology by Beilinson) denotes the \(k^ j\)-eigenspace of \(K_ i(S) \otimes \mathbb{Q}\) under the Adams operations \(\psi_ k\), and where the extensions are taken in the abelian category of \(\mathbb{Q}\)- (or \(\mathbb{R}\)-) mixed Hodge structures on \(S(\mathbb{C})\) (and may be identified with Deligne-Beilinson (or absolute Hodge) cohomology). In particular, if \(S = \text{Spec} (F)\) for a number field \(F\), one gets for every complex embedding \(\sigma\) of \(F\) a regulator map \[ \text{reg}^ \sigma : K_{2k-1} (F) \to \text{Ext}^ 1_{S(\mathbb{C})} \bigl( \mathbb{Q} (0), \mathbb{Q} (k ) \bigr) = \mathbb{C}/(2 \pi i)^ k \mathbb{Q} \] which does not depend on the choice of \(i\), thus \(\text{reg}^{\overline \sigma} (x) = \text{reg}^ \sigma (x)^ -\). Passing over to \(\mathbb{R}\)-Hodge structures one gets, for every complex embedding \(\sigma\) of \(F\), \[ \text{reg}^ \sigma_ \mathbb{R} : K_{2k-1} (F) \to \mathbb{C}/ (2 \pi i)^ k \mathbb{R} @> \sim>> i^{k-1} \mathbb{R} \] such that \(\text{reg}_ \mathbb{R}^{\overline \sigma} (x) = \text{reg}^ \sigma_ \mathbb{R}(x)^ -\). Zagier’s conjecture relates this regulator to his higher polylogarithms \(D_ k\): If \(Li_ k(z)\), \(z \in \mathbb{P}^ 1 (\mathbb{C}) \setminus \{0,1,\infty\}\), denotes the multivalued analytic continuation of the series \(Li_ k (z) = \sum_{n = 1}^ \infty {z^ n \over n^ k}\), \(| z | < 1\) and \(k \geq 1\), Zagier’s polylogarithm \(D_ k (z)\) is defined as \(D_ k(z) = \text{Re}_ k (\sum_{\ell = 0}^{k-1} b_ \ell {\log (z \overline z)^ \ell \over \ell!} Li_{k - \ell}(z))\), where \(\text{Re}_ k = \text{Re}e\) when \(k\) is odd and \(\text{Re}_ k = i \text{Im} m\) when \(k\) is even. The \(b_ \ell\) are the Bernoulli numbers. The \(D_ k\) are real single-valued functions, real analytic on \(\mathbb{P}^ 1(\mathbb{C}) \setminus \{0,1,\infty\}\). They are continuous on all of \(\mathbb{P}^ 1(\mathbb{C})\) and \(D_ k\) is \((-1)^{k-1}\)-symmetric with respect to \(z \mapsto 1/z\) or \(z \mapsto \overline z\). Furthermore, they (should) satisfy ‘clean’ functional equations.
Zagier’s conjecture is now an iterated system of conjectures \((C_ k)\), \(k = 1,2, \dots\), where in \((C_ k)\) one defines a \(\mathbb{Q}\)-vector space \({\mathcal L}^ k\), a map \(\{ \}_ k : F \to {\mathcal L}^ k\), a homomorphism \(d_ k : {\mathcal L}^ k \to \wedge^ 2 (\oplus_{\ell=1}^{k-1} {\mathcal L}^ \ell)\) and \(\varphi_ k : \text{Ker} (d_ k) \hookrightarrow K_{2k-1} (F) \otimes \mathbb{Q}\), with \({\mathcal L}^ 1 = F^ \times \otimes \mathbb{Q}\), \(\{ \}_ 1 : x \mapsto (1-x)\) where \((x)\) denotes the image of \(x \in F^ \times\) in \({\mathcal L}^ 1\), \(d_ 1 = 0\) and \(\varphi_ 1\) is the identity on \(F^ \times \otimes \mathbb{Q}\). For \(k \geq 2\) one defines the \(\mathbb{Q}\)-vector space \(\widetilde {\mathcal L}^ k\) freely generated by symbols \(\{x\}_ k^ \sim\), \(x \in F - \{0,1\}\), and \[ \widetilde d_ k : \widetilde {\mathcal L}^ k \to {\mathcal L}^{k - 1} \otimes {\mathcal L}^ 1 \to \wedge^ 2 \Bigl( \oplus_{\ell = 1}^{k-1} {\mathcal L}^ \ell \Bigr), \] given by \(\{x\}_ k^ \sim \mapsto \{x\}_ k \otimes (x) \mapsto \{x\}_ k \wedge(x)\). The conjecture now states the existence of a map \[ \widetilde \varphi_ k : \text{Ker} \bigl( \widetilde d_ k : \widetilde {\mathcal L}^ k \to \wedge^ 2 (\oplus_{\ell = 1}^{k-1} {\mathcal L}^ \ell) \bigr) \to K_{2k - 1} (F) \otimes \mathbb{Q} \] such that for every complex embedding \(\sigma\) of \(F\) and any \(x = \sum \lambda_ \alpha \{x_ \alpha\}_ k^ \sim \in \text{Ker} (\widetilde d_ k)\), one has \(D_ k (\sigma x) : = \sum \lambda_ \alpha D_ k (\sigma x_ \alpha) = - \text{reg}^ \sigma_ \mathbb{R} (\widetilde \varphi_ k(x))\). If this holds one defines \({\mathcal L}^ k : = \widetilde {\mathcal L}^ k/ \text{Ker} (\widetilde \varphi_ k)\), and \(\{ \}\), \(d_ k\) and \(\varphi_ k\) are defined by passing to the quotient. Zagier conjectures \(\varphi_ k\) to be surjective. This point is not discussed in the underlying paper. If \({\mathcal L}\) denotes the sum of the \({\mathcal L}^ i\), then \(({\mathcal L},d)\) becomes a graded Lie co-algebra.
The aim of the underlying article is to show that Zagier’s conjecture follows from the existence of a suitable tannakian \(\mathbb{Q}\)-linear category \({\mathcal T} (S)\) of so-called mixed Tate motives over \(S\). For varying \(S\) the \({\mathcal T} (S)\) should be a stack. Every \({\mathcal T} (S)\) should contain a fixed invertible object of rank 1, the Tate object \(\mathbb{Q} (1)\), such that any simple object of \({\mathcal T} (S)\) is isomorphic to some tensor power of \(\mathbb{Q} (1)\). One should have that \(\operatorname{Hom}_{{\mathcal T} (S)} (\mathbb{Q}(0), \mathbb{Q} (k)) = \mathbb{Q}\) for \(k = 0\) and =0 otherwise, and \(\text{Ext}^ 1_{{\mathcal T} (S)} (\mathbb{Q} (0), \mathbb{Q}(k)) = 0\) for \(k \leq 0\). For \(k \geq 1\) one should have an isomorphism \(\alpha : (K_{2k-1} (S) \otimes \mathbb{Q})^{(k)} @> \sim>>\text{Ext}^ 1_{{\mathcal T} (S)} (\mathbb{Q} (0), \mathbb{Q} (k))\). This amounts to the existence of a unique finite increasing filtration \(W\) of every object \(M\) of \({\mathcal T} (S)\), which for convenience may be indexed by even integers, such that \(\text{Gr}^ W_{ - 2k} (M)\) is a sum of copies of \(\mathbb{Q}(k)\). The functor \(M \mapsto \text{Gr}^ W(M)\) is exact.
There should be a tensor functor ‘real’, called realization, from \({\mathcal T} (S)\) to the category of variations of \(\mathbb{Q}\)-mixed Hodge structures on \(S(C)\), for an algebraic closure \(C\) of \(\mathbb{R}\). This realization should be compatible with base change and functorial in \(C\). The Tate motive \(\mathbb{Q}(1)\) should be mapped to the Tate Hodge structure \(\mathbb{Q}(1)\). The regulator map is related by the realization via \(\alpha\): reg = real\(\circ \alpha\). For \(S = \mathbb{P}^ 1 \setminus \{0,1, \infty\}\) one demands the existence of a projective system of extensions \(_ 0{\mathcal M}^{(N)}\) of \(\mathbb{Q}(0)\) by the \(\text{Sym}^{(N-1)} ([z]) (1)\), where for \(f \in {\mathcal O}^ \times (S)\), (the realization of) \([f]\) corresponds to the extension of \(\mathbb{Q}(0)\) by \(\mathbb{Q} (1)\) given by the matrix \((\begin{matrix} 1 & 0 \\ \log(f) & 2 \pi i \end{matrix})\). By definition, \(_ 0{\mathcal M}^{(1)} = [1-z]\). One can identify \(\text{Gr}^ W \text{Sym}^{N-1} ([z])(1)\) with \(\oplus_ 1^ N \mathbb{Q}(k)\). As a matter of fact, one can do with a slightly weaker form of these conjectures to deal with Zagier’s conjecture, in particular, one may take \(S = \mathbb{P}^ 1 \setminus \{0,1,\infty\}\) or \(S = \text{Spec} (F)\). One writes \({\mathcal T} (F)\) for \({\mathcal T} (\text{Spec} (F))\).
The paper is divided in four sections. In the first one an explicit variation of \(\mathbb{Q}\)- (or \(\mathbb{R}\)-) mixed Hodge structures \({\mathcal M}^{(N)}\) over \(S = \mathbb{P}^ 1 \setminus \{0,1,\infty\}\), determined by \(\log (z)\) and the \(Li_ k(z)\) \((k = 1,\dots,N)\), is constructed such that the only nonzero Hodge numbers are the \(h^{pp}\), \(p \in \mathbb{Z}\). Such variations are called mixed Tate variations. The \({\mathcal M}^{(N)}\) do not depend on the choice of \(i \in \mathbb{C})\) and they form a projective system in \(N\). They can be shown to be the Hodge components of a system of realizations as defined in the motivic formalism of the fundamental group \(\mathbb{P}^ 1 \setminus \{0,1, \infty\}\). Moreover it is shown that the actual real mixed Hodge structure \({\mathcal M}^{(N)}_ z\), \(z \in \mathbb{P}^ 1 (\mathbb{C}) \setminus \{0,1,\infty\}\), is characterized by Zagier’s polylogarithms \(D_ k(z)\), \(k=1, \dots,N\). In the second section the tannakian formalism is applied to construct the \(\{z\}_ k\) as elements of the \(k\)-component (Lie \(U^ \vee)^ k\) of a graded Lie co-algebra Lie \(U^ \vee\) determined by a fibre functor of the tannakian category \({\mathcal T} (S)\). The main result can then be formulated as follows: There exists a monomorphism of Lie co-algebras \(\varphi : {\mathcal L} \to (\text{Lie} U)^ \vee\), compatible with \(\{ \}_ k\), such that Zagier’s conjecture holds for \(\varphi_ k\) defined as the composition \[ \text{Ker} \bigl (\widetilde d_ k : \widetilde {\mathcal L}^ k \to \wedge^ 2 \oplus_ 1^{k-1} {\mathcal L}^ \ell \bigr) @>\varphi>> \text{Ker} \bigl( d : \text{Lie} U^ \vee \to \wedge^ 2 \text{Lie } U^ \vee \bigr)^ k, \] where this last \(\text{Ker} (d)\) can be shown to be equal to \(\text{Ext}^ 1_{\mathcal T} (F) (\mathbb{Q} (0),\mathbb{Q} (k)) = K_{2k-1} (F) \otimes \mathbb{Q}\) (according to hypothesis). The \(\varphi_ k\) are inductively defined via \(\widetilde \varphi_ k : \widetilde {\mathcal L}^ k \to (\text{Lie} U^ \vee)^ k\), \(\{z\}_ k^ \sim \mapsto \{z\}_ k\).
The fibre functor \(\omega\) mentioned before takes its values in the graded \(\mathbb{Q}\)-vector spaces and is defined by \(\omega (M)_ k = \operatorname{Hom}_{{\mathcal T} (S)} (\mathbb{Q} (k), \text{Gr}^ W_{-2k} (M))\), and \(\omega (M) = \oplus \omega (M)_ k\). The category \({\mathcal T} = {\mathcal T} (S)\) is equivalent to the category of representations of the group scheme \(G = \underline {\operatorname{Aut}}^ \otimes (\omega)\) and the grading implies that \(G\) is the semi-direct product of \(\mathbb{G}_ m\) with a pro-unipotent group scheme \(U\); actually, \(G\) is the projective limit of \(\mathbb{G}_ m\) with the system of unipotent algebraic groups \(U_ \alpha\). Passing to the duals and taking the inductive limit one obtains a graded Lie co-algebra \(((\text{Lie} U^ \vee),d)\) with the crucial property that an \(e \in (\text{Lie} U^ \vee )^ k\) with \(de = 0\) corresponds to an extension of \(\mathbb{Q}(0)\) with \(\mathbb{Q}(k)\). For \(M \in {\mathcal T}\), \(x \in \omega (M)_ i\), \(y \in \omega (M)_ j^ \vee)\) and \(k=j-i\) one defines a linear form \(c_{y, x} \in (\text{Lie} U^ \vee)^ k\) by \(c_{y,x} (u) = \langle y,ux \rangle\), \(u \in \text{Lie} U_ \alpha\), and the essential idea is that a linear combination of such coefficients that vanish under \(d\) giv an extension of \(\mathbb{Q}(0)\) by \(\mathbb{Q}(k)\). Then one defines \(\{z\}_ k : = c_{y,x} \in (\text{Lie} U^ \vee)^ k\), and it is shown that \(\{z\}_ 1 = (1-z)\) and \(d\{z\}_ k = \{z\}_{k-1} \wedge (z)\) in \(\wedge^ 2 \text{Lie} U^ \vee\). Now a complex embedding \(\sigma\) of \(F\) into \(\mathbb{C}\) leads to a realization functor ‘real’ of \({\mathcal T} (F)\) into \(\mathbb{Q}\)-mixed Tate structures, i.e., \(\mathbb{Q}\)-mixed Hodge structures with only \(h^{pp} \neq 0\). Going over to the underlying real structure, one gets a realization functor ‘real\(_ \mathbb{R}\)’ with values in the tannakian category of \(\mathbb{R}\)-mixed Tate structures. The description of such structures leads to a \(\otimes\)-equivalence of the category of \(\mathbb{R}\)-mixed Tate structures with the category of real graded vector spaces equipped with elements \(N_ k \in \text{GL} (\omega_ \mathbb{R}) \otimes \mathbb{C}\), \(k \geq 1\), of degree \(k\) and with \(N_ k\) real for odd \(k\) and purely imaginary for even \(k\). For \(M = {\mathcal M}_ z\), \(z \in F \setminus \{0\}\), one obtains the result \(\langle N_ k\), \(\{z\}_ k \rangle = -D_ k (z)\). Finally, one observes that, for an extension \(E\) of \(\mathbb{Q}(0)\) by \(\mathbb{Q}(k)\) defining \(x \in \omega(E)_ 0\) and \(y \in \omega (E)_ k^ \vee\), one has the fact that \(\langle N_ k,\;c_{y,x} \rangle\) is the regulator \(\text{reg}^ \sigma_ \mathbb{R}\) of the corresponding class in \(K_{2k - 1} (F) \otimes \mathbb{Q}\). In the third section the principle of using the linear forms \(c_{y,x}\), in particular their origin as derived from coefficients of the representation of \(U\), is explicitly discussed. In the final section it is shown how Zagier’s conjecture can be interpreted as the real form of a complex conjecture that is also implied by the motivic formalism.
For the entire collection see [Zbl 0788.00054].

MSC:
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
33E20 Other functions defined by series and integrals
14A20 Generalizations (algebraic spaces, stacks)
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