Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields.

*(English)*Zbl 0728.11062
Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 391-430 (1991).

[For the entire collection see Zbl 0711.00011.]

Let F be an algebraic number field with \(r_ 1\) real and \(r_ 2\) (pairs of) complex embeddings, and write \(n_+=r_ 1+r_ 2\) and \(n_ -=r_ 2\). \(\Delta\) will denote the absolute value of the discriminant of F and \(\zeta_ F(s)\) its Dedekind zeta function. Furthermore, let \(Li_ k(x)=\sum^{\infty}_{n=1}\frac{x^ n}{n^ k}\), \(k\in {\mathbb{N}}\), \(| x| \leq 1\) and by analytic continuation if \(| x| >1\), be the kth polylogarithm function. The aim of the underlying paper is to relate \(\zeta_ F(s)\) at \(s=m\), \(m=2,3,...\), to special values of a somewhat modified form \(P_ m(x)\) (or \(D_ m(x))\) of the \(Li_ m(x)\). At the same time one is led to the discovery of functional equations of the polylogarithms. To ‘clean’ these functional equations, i.e. to eliminate lower-order polylogarithms or ordinary logarithms, a good choice for \(P_ m(x)\) is \[ P_ m(x)=\quad {\mathfrak R}_ m(\sum^{m}_{j=0}\frac{2^ jB_ j}{j!}(\log | x|)^ jLi_{m-j}(x)), \] where \({\mathfrak R}_ m\) denotes \({\mathfrak R}\) or \({\mathfrak I}\) according to whether m is odd or even, respectively. The \(B_ j's\) are the Bernoulli numbers. The \(P_ m(x)\) are one-valued, real analytic on \({\mathbb{P}}^ 1({\mathbb{C}})\setminus \{0,1,\infty \}\), continuous on all of \({\mathbb{P}}^ 1({\mathbb{C}})\), and \((-1)^{m-1}\)-symmetric with respect to \(x\mapsto 1/x\) or \(x\mapsto \bar x\). The \(D_ m\) and the \(P_ m\) are related by \[ D_ m(x)=\sum_{0\leq 2j<m}\frac{\log^{2j}| x|}{(2j+1)!}P_{m-2j}(x). \] For each embedding \(\sigma: F\to {\mathbb{C}}\) one defines \(P^{\sigma}_ m\) as the composite of \(\sigma\) with the map \(P_ m: {\mathbb{P}}^ 1({\mathbb{C}})\to {\mathbb{R}}\) and then \(P^ F_ m: {\mathcal F}_ F\to {\mathbb{R}}^{n\mp}(where\) \((-1)^ m=\pm)\) as the product of the \(P^{\sigma}_ m\) over all the real and half the complex embeddings of F (one of each complex conjugate pair) if m is odd and over half the complex embeddings if m is even. Here \({\mathcal F}_ F\) is the free abelian group on \(F^{\times}\) with elements the finite linear combinations \(\sum_ in_ i[x_ i]\), \(n_ i\in {\mathbb{Z}}\), \(x_ i\in F^{\times}\). \({\mathcal F}_ F\) may be identified with the quotient of the free abelian group on \({\mathbb{P}}^ 1(F)\) by the subgroup \(<[0],[\infty \}>\). By definition, \(P^ F_ m(\sum_ in_ i[x_ i])=(\sum_ in_ iP_ m(\sigma (x_ i)))_{\sigma}\). Actually, one constructs the mth Bloch group \({\mathcal B}_ m(F)\) of F as a quotient of a subgroup \({\mathcal A}_ m(F)\) of \({\mathcal F}_ F\) by a subgroup \({\mathcal C}_ m(F)\) reflecting the functional equations satisfied by \(P_ m(x)\), to get a homomorphism \(P^ F_ m: {\mathcal B}_ m(F)\to {\mathbb{R}}^{n\mp}\). Then one can state, motivated by a large number of numerical examples (mostly for \(m=3)\), the Main Conjecture in the following form:

Main Conjecture 1. \(P^ F_ m\) is an isomorphism from \({\mathcal B}_ m(F)\) onto a lattice in \({\mathbb{R}}^{n\mp}\) whose volume is a rational multiple of \(\sqrt{\Delta}\zeta_ F(m)/\pi^{mn\mp}.\)

For cyclotomic F it is proved that the subgroup of \({\mathcal B}_ m(F)\) generated by the roots of unity in F is mapped under \(P^ F_ m\) onto a lattice in \({\mathbb{R}}^{n\mp}\) with volume as described in the conjecture. There is also an analogous conjecture for \(D^ F_ m\). As a matter of fact, one would like to relate the lattice alluded to above to the one constructed via Borel’s regulator map \(K_{2m-1}(F)\to {\mathbb{R}}^{n\mp},\) where the \(K_ i(F)\) are the higher algebraic K- groups of F. The final formulation of the Main Conjecture becomes:

Main Conjecture 2. There is a canonical map \({\mathcal B}_ m(F)\to K_{2m- 1}(F)\) with finite kernel and cokernel, whose composite with Borel’s regulator map coincides with \(P^ F_ m.\)

For \(m=2\), \(P_ 2\) is the Bloch-Wigner function, i.e. the ‘cleaned’ dilogarithm, and the conjectures were proven by the author and by Bloch and Suslin. For the first conjecture \(P_ 2(x)\) is interpreted as a hyperbolic volume. The triangulation of a complete hyperbolic 3-manifold then suggests the definition of \({\mathcal A}(F)={\mathcal A}_ 2(F)\) as the subgroup of elements \(\sum_ in_ i[x_ i]\) of \({\mathcal F}_ F\) such that \(\sum_ in_ i[x_ i]\wedge [1-x_ i]=0\) in \(\wedge^ 2(F^{\times})\otimes_{{\mathbb{Z}}}{\mathbb{Q}},\quad x_ i\neq 1.\) Using the “five-term relation” of Spence and Abel for the dilogarithm, one takes \({\mathcal C}(F)={\mathcal C}_ 2(F)=<S_{xy}>\), where \[ S_{xy}=[x]+[y]+[\frac{1-x}{1-xy}]+[1-xy]+[\frac{1-y}{1-xy}],\quad x,y\in {\mathbb{P}}^ 1(F),\quad (x,y)\neq (0,\infty),\quad (\infty,0),\quad (1,1). \] The (second) Bloch group \({\mathcal B}(F)={\mathcal B}_ 2(F)\) is now \({\mathcal A}(F)/{\mathcal C}(F)\). The main motivation for the various forms of the Main Conjecture comes from high precision computer computations expressing \(\sqrt{D}\zeta_{{\mathbb{Q}}(\sqrt{D})}(3)\), \(D>0\), as a rational linear combination of \(D_ 3(x)'s\) for ‘simple’ numbers \(x\in {\mathbb{Q}}(\sqrt{D})\), i.e. numbers x such that both x and 1-x factor into prime ideals of small norm, and by using the Lenstra-Lenstra-Lovasz algorithm. Based on these numerical examples, the search for \({\mathcal A}_ 3(F)\) leads to consider the group \[ {\mathcal A}_ 3^{(3)}(F)=Ker(\beta_ 3: {\mathcal F}_ F\to (F^{\times}\otimes \wedge^ 2(F^{\times}))\otimes {\mathbb{Q}}), \] with \(\beta_ 3:\sum n_ i[x_ i]\mapsto \sum n_ i[x_ i]\otimes ([x_ i]\wedge [1-x_ i])\), [1]\(\mapsto 0\). For any linear map v: \(F^{\times}\to {\mathbb{Z}}\) let \(\iota_ v: {\mathcal F}_ F\to {\mathcal F}_ F\) be defined on generators by \(\iota_ v: [x]\mapsto v(x)[x]\). Then \(\iota_ v\) maps \({\mathcal A}_ 3^{(3)}(F)\) to \({\mathcal A}(F)={\mathcal A}_ 2(F)\). \({\mathcal A}_ 3(F)\) is now defined as the subgroup of \({\mathcal A}_ 3^{(3)}(F)\) that maps to \({\mathcal C}_ 2(F)\) for all v: \(F^{\times}\to {\mathbb{Z}}\). For \({\mathcal C}_ 3(F)\) one is led to take the group of the form \(<S_{xy}^{(3)}>\), where \(S_{xy}^{(3)}\) is the 2-variable relation for the trilogarithm found by Spence (1809) and Kummer (1840). One speculates that all functional equations for the trilogarithm follow from the Spence-Kummer ones. The (third) Bloch group \({\mathcal B}_ 3(F)\) is \({\mathcal A}_ 3(F)/{\mathcal C}_ 3(F)\). The general case is based on a remarkable proposition:

Proposition. Let \(\{n_ i,x_ i(t)\}\) be a collection of integers \(n_ i\) and rational functions of one variable \(x_ i(t)\) satisfying the identity \[ \sum_{i}n_ i[x_ i(t)]^{m-2}\otimes ([x_ i(t)]\wedge [1-x_ i(t)])=0\text{ in } (Sym^{m-2}({\mathbb{C}}(t)^{\times})\otimes \wedge^ 2({\mathbb{C}}(t)^{\times}))\otimes_{{\mathbb{Z}}}{\mathbb{Q}}. \] Then \(\sum_ in_ iP_ m(x_ i(t))=text{constant}.\)

The \({\mathcal A}_ m(F)\) and \({\mathcal C}_ m(F)\) are then defined recursively as follows:

1. \({\mathcal A}_ m^{(3)}(F)\subset {\mathcal F}_ F\) is defined as Ker \((\beta_ m)\) with \(\beta_ m: {\mathcal F}_ F\to Sym^{m- 2}(F^{\times}_{{\mathbb{Q}}})\otimes \wedge^ 2(F^{\times}_{{\mathbb{Q}}})\) \((F^{\times}_{{\mathbb{Q}}}=F^{\times}\otimes_{{\mathbb{Z}}}{\mathbb{Q}})\) given on generators by \(\beta_ m: [x]\mapsto [x]^{m-2}\otimes ([x]\wedge [1-x]),\) \(x\in {\mathbb{P}}^ 1(F)\setminus \{0,1,\infty \}\), and \(\beta_ m([x])=0\) for \(x\in \{0,1,\infty \}.\)

2. \({\mathcal C}_ m(F)\subset {\mathcal A}_ m^{(3)}(F)\) is the subgroup generated by all images \((\phi_{\alpha}-\phi_{\beta})({\mathcal A}_ m^{(3)}({\mathbb{Q}}(t)))\), \(\alpha,\beta \in {\mathbb{P}}^ 1(F)\), where \(\phi_{\alpha}:{\mathcal F}_{{\mathbb{Q}}(t)}\to {\mathcal F}_ F\) is the evaluation map defined on generators by [x(t)]\(\mapsto [x(\alpha)]\). \({\mathcal C}_ m(F)\) should be interpreted as the group spanned by the functional equations (which are unknown explicitly, in general) of \(P_ m.\)

3. With \(\iota: {\mathcal F}_ F\to F^{\times}\otimes {\mathcal F}_ F\), [x]\(\mapsto [x]\otimes [x]\), one sets \({\mathcal A}_ m(F)=\iota^{- 1}(F^{\times}\otimes {\mathcal C}_{m-1}(F))\) and defines the m-th Bloch group \({\mathcal B}_ m(F)\) as \({\mathcal A}_ m(F)/{\mathcal C}_ m(F).\)

Apart from the result for cyclotomic fields and the motivating examples for the trilogarithm, a number of numerical examples supporting the Main Conjecture are given: relations for the pentalogarithm for rational arguments, a 30-term identity for the heptalogarithm (in collaboration with H. Gangl) of suitable rational numbers, and “ladder relations” of L. Lewin et al.. The paper closes with a penultimate section of complements (on: the conditions defining \({\mathcal A}_ m(F)\), rational independence of polylogarithm values, cf. Milnor’s conjecture for the dilogarithm, and generalization of the whole situation to Artin L- functions).

The final section describes some recent developments, one of the most stimulating being A. Goncharov’s proof of the conjectures for \(m=3\). Also, recent work of P. Deligne and of A. Beilinson is briefly reported. Beilinson’s work combined with the results of the penultimate section implies e.g. that there are infinitely many relations over \({\mathbb{Q}}\) among the values of the polylogarithm function of arbitrary order at algebraic, or even rational, arguments.

Let F be an algebraic number field with \(r_ 1\) real and \(r_ 2\) (pairs of) complex embeddings, and write \(n_+=r_ 1+r_ 2\) and \(n_ -=r_ 2\). \(\Delta\) will denote the absolute value of the discriminant of F and \(\zeta_ F(s)\) its Dedekind zeta function. Furthermore, let \(Li_ k(x)=\sum^{\infty}_{n=1}\frac{x^ n}{n^ k}\), \(k\in {\mathbb{N}}\), \(| x| \leq 1\) and by analytic continuation if \(| x| >1\), be the kth polylogarithm function. The aim of the underlying paper is to relate \(\zeta_ F(s)\) at \(s=m\), \(m=2,3,...\), to special values of a somewhat modified form \(P_ m(x)\) (or \(D_ m(x))\) of the \(Li_ m(x)\). At the same time one is led to the discovery of functional equations of the polylogarithms. To ‘clean’ these functional equations, i.e. to eliminate lower-order polylogarithms or ordinary logarithms, a good choice for \(P_ m(x)\) is \[ P_ m(x)=\quad {\mathfrak R}_ m(\sum^{m}_{j=0}\frac{2^ jB_ j}{j!}(\log | x|)^ jLi_{m-j}(x)), \] where \({\mathfrak R}_ m\) denotes \({\mathfrak R}\) or \({\mathfrak I}\) according to whether m is odd or even, respectively. The \(B_ j's\) are the Bernoulli numbers. The \(P_ m(x)\) are one-valued, real analytic on \({\mathbb{P}}^ 1({\mathbb{C}})\setminus \{0,1,\infty \}\), continuous on all of \({\mathbb{P}}^ 1({\mathbb{C}})\), and \((-1)^{m-1}\)-symmetric with respect to \(x\mapsto 1/x\) or \(x\mapsto \bar x\). The \(D_ m\) and the \(P_ m\) are related by \[ D_ m(x)=\sum_{0\leq 2j<m}\frac{\log^{2j}| x|}{(2j+1)!}P_{m-2j}(x). \] For each embedding \(\sigma: F\to {\mathbb{C}}\) one defines \(P^{\sigma}_ m\) as the composite of \(\sigma\) with the map \(P_ m: {\mathbb{P}}^ 1({\mathbb{C}})\to {\mathbb{R}}\) and then \(P^ F_ m: {\mathcal F}_ F\to {\mathbb{R}}^{n\mp}(where\) \((-1)^ m=\pm)\) as the product of the \(P^{\sigma}_ m\) over all the real and half the complex embeddings of F (one of each complex conjugate pair) if m is odd and over half the complex embeddings if m is even. Here \({\mathcal F}_ F\) is the free abelian group on \(F^{\times}\) with elements the finite linear combinations \(\sum_ in_ i[x_ i]\), \(n_ i\in {\mathbb{Z}}\), \(x_ i\in F^{\times}\). \({\mathcal F}_ F\) may be identified with the quotient of the free abelian group on \({\mathbb{P}}^ 1(F)\) by the subgroup \(<[0],[\infty \}>\). By definition, \(P^ F_ m(\sum_ in_ i[x_ i])=(\sum_ in_ iP_ m(\sigma (x_ i)))_{\sigma}\). Actually, one constructs the mth Bloch group \({\mathcal B}_ m(F)\) of F as a quotient of a subgroup \({\mathcal A}_ m(F)\) of \({\mathcal F}_ F\) by a subgroup \({\mathcal C}_ m(F)\) reflecting the functional equations satisfied by \(P_ m(x)\), to get a homomorphism \(P^ F_ m: {\mathcal B}_ m(F)\to {\mathbb{R}}^{n\mp}\). Then one can state, motivated by a large number of numerical examples (mostly for \(m=3)\), the Main Conjecture in the following form:

Main Conjecture 1. \(P^ F_ m\) is an isomorphism from \({\mathcal B}_ m(F)\) onto a lattice in \({\mathbb{R}}^{n\mp}\) whose volume is a rational multiple of \(\sqrt{\Delta}\zeta_ F(m)/\pi^{mn\mp}.\)

For cyclotomic F it is proved that the subgroup of \({\mathcal B}_ m(F)\) generated by the roots of unity in F is mapped under \(P^ F_ m\) onto a lattice in \({\mathbb{R}}^{n\mp}\) with volume as described in the conjecture. There is also an analogous conjecture for \(D^ F_ m\). As a matter of fact, one would like to relate the lattice alluded to above to the one constructed via Borel’s regulator map \(K_{2m-1}(F)\to {\mathbb{R}}^{n\mp},\) where the \(K_ i(F)\) are the higher algebraic K- groups of F. The final formulation of the Main Conjecture becomes:

Main Conjecture 2. There is a canonical map \({\mathcal B}_ m(F)\to K_{2m- 1}(F)\) with finite kernel and cokernel, whose composite with Borel’s regulator map coincides with \(P^ F_ m.\)

For \(m=2\), \(P_ 2\) is the Bloch-Wigner function, i.e. the ‘cleaned’ dilogarithm, and the conjectures were proven by the author and by Bloch and Suslin. For the first conjecture \(P_ 2(x)\) is interpreted as a hyperbolic volume. The triangulation of a complete hyperbolic 3-manifold then suggests the definition of \({\mathcal A}(F)={\mathcal A}_ 2(F)\) as the subgroup of elements \(\sum_ in_ i[x_ i]\) of \({\mathcal F}_ F\) such that \(\sum_ in_ i[x_ i]\wedge [1-x_ i]=0\) in \(\wedge^ 2(F^{\times})\otimes_{{\mathbb{Z}}}{\mathbb{Q}},\quad x_ i\neq 1.\) Using the “five-term relation” of Spence and Abel for the dilogarithm, one takes \({\mathcal C}(F)={\mathcal C}_ 2(F)=<S_{xy}>\), where \[ S_{xy}=[x]+[y]+[\frac{1-x}{1-xy}]+[1-xy]+[\frac{1-y}{1-xy}],\quad x,y\in {\mathbb{P}}^ 1(F),\quad (x,y)\neq (0,\infty),\quad (\infty,0),\quad (1,1). \] The (second) Bloch group \({\mathcal B}(F)={\mathcal B}_ 2(F)\) is now \({\mathcal A}(F)/{\mathcal C}(F)\). The main motivation for the various forms of the Main Conjecture comes from high precision computer computations expressing \(\sqrt{D}\zeta_{{\mathbb{Q}}(\sqrt{D})}(3)\), \(D>0\), as a rational linear combination of \(D_ 3(x)'s\) for ‘simple’ numbers \(x\in {\mathbb{Q}}(\sqrt{D})\), i.e. numbers x such that both x and 1-x factor into prime ideals of small norm, and by using the Lenstra-Lenstra-Lovasz algorithm. Based on these numerical examples, the search for \({\mathcal A}_ 3(F)\) leads to consider the group \[ {\mathcal A}_ 3^{(3)}(F)=Ker(\beta_ 3: {\mathcal F}_ F\to (F^{\times}\otimes \wedge^ 2(F^{\times}))\otimes {\mathbb{Q}}), \] with \(\beta_ 3:\sum n_ i[x_ i]\mapsto \sum n_ i[x_ i]\otimes ([x_ i]\wedge [1-x_ i])\), [1]\(\mapsto 0\). For any linear map v: \(F^{\times}\to {\mathbb{Z}}\) let \(\iota_ v: {\mathcal F}_ F\to {\mathcal F}_ F\) be defined on generators by \(\iota_ v: [x]\mapsto v(x)[x]\). Then \(\iota_ v\) maps \({\mathcal A}_ 3^{(3)}(F)\) to \({\mathcal A}(F)={\mathcal A}_ 2(F)\). \({\mathcal A}_ 3(F)\) is now defined as the subgroup of \({\mathcal A}_ 3^{(3)}(F)\) that maps to \({\mathcal C}_ 2(F)\) for all v: \(F^{\times}\to {\mathbb{Z}}\). For \({\mathcal C}_ 3(F)\) one is led to take the group of the form \(<S_{xy}^{(3)}>\), where \(S_{xy}^{(3)}\) is the 2-variable relation for the trilogarithm found by Spence (1809) and Kummer (1840). One speculates that all functional equations for the trilogarithm follow from the Spence-Kummer ones. The (third) Bloch group \({\mathcal B}_ 3(F)\) is \({\mathcal A}_ 3(F)/{\mathcal C}_ 3(F)\). The general case is based on a remarkable proposition:

Proposition. Let \(\{n_ i,x_ i(t)\}\) be a collection of integers \(n_ i\) and rational functions of one variable \(x_ i(t)\) satisfying the identity \[ \sum_{i}n_ i[x_ i(t)]^{m-2}\otimes ([x_ i(t)]\wedge [1-x_ i(t)])=0\text{ in } (Sym^{m-2}({\mathbb{C}}(t)^{\times})\otimes \wedge^ 2({\mathbb{C}}(t)^{\times}))\otimes_{{\mathbb{Z}}}{\mathbb{Q}}. \] Then \(\sum_ in_ iP_ m(x_ i(t))=text{constant}.\)

The \({\mathcal A}_ m(F)\) and \({\mathcal C}_ m(F)\) are then defined recursively as follows:

1. \({\mathcal A}_ m^{(3)}(F)\subset {\mathcal F}_ F\) is defined as Ker \((\beta_ m)\) with \(\beta_ m: {\mathcal F}_ F\to Sym^{m- 2}(F^{\times}_{{\mathbb{Q}}})\otimes \wedge^ 2(F^{\times}_{{\mathbb{Q}}})\) \((F^{\times}_{{\mathbb{Q}}}=F^{\times}\otimes_{{\mathbb{Z}}}{\mathbb{Q}})\) given on generators by \(\beta_ m: [x]\mapsto [x]^{m-2}\otimes ([x]\wedge [1-x]),\) \(x\in {\mathbb{P}}^ 1(F)\setminus \{0,1,\infty \}\), and \(\beta_ m([x])=0\) for \(x\in \{0,1,\infty \}.\)

2. \({\mathcal C}_ m(F)\subset {\mathcal A}_ m^{(3)}(F)\) is the subgroup generated by all images \((\phi_{\alpha}-\phi_{\beta})({\mathcal A}_ m^{(3)}({\mathbb{Q}}(t)))\), \(\alpha,\beta \in {\mathbb{P}}^ 1(F)\), where \(\phi_{\alpha}:{\mathcal F}_{{\mathbb{Q}}(t)}\to {\mathcal F}_ F\) is the evaluation map defined on generators by [x(t)]\(\mapsto [x(\alpha)]\). \({\mathcal C}_ m(F)\) should be interpreted as the group spanned by the functional equations (which are unknown explicitly, in general) of \(P_ m.\)

3. With \(\iota: {\mathcal F}_ F\to F^{\times}\otimes {\mathcal F}_ F\), [x]\(\mapsto [x]\otimes [x]\), one sets \({\mathcal A}_ m(F)=\iota^{- 1}(F^{\times}\otimes {\mathcal C}_{m-1}(F))\) and defines the m-th Bloch group \({\mathcal B}_ m(F)\) as \({\mathcal A}_ m(F)/{\mathcal C}_ m(F).\)

Apart from the result for cyclotomic fields and the motivating examples for the trilogarithm, a number of numerical examples supporting the Main Conjecture are given: relations for the pentalogarithm for rational arguments, a 30-term identity for the heptalogarithm (in collaboration with H. Gangl) of suitable rational numbers, and “ladder relations” of L. Lewin et al.. The paper closes with a penultimate section of complements (on: the conditions defining \({\mathcal A}_ m(F)\), rational independence of polylogarithm values, cf. Milnor’s conjecture for the dilogarithm, and generalization of the whole situation to Artin L- functions).

The final section describes some recent developments, one of the most stimulating being A. Goncharov’s proof of the conjectures for \(m=3\). Also, recent work of P. Deligne and of A. Beilinson is briefly reported. Beilinson’s work combined with the results of the penultimate section implies e.g. that there are infinitely many relations over \({\mathbb{Q}}\) among the values of the polylogarithm function of arbitrary order at algebraic, or even rational, arguments.

Reviewer: W.W.J.Hulsbergen (Breda)

##### MSC:

11R42 | Zeta functions and \(L\)-functions of number fields |

11R70 | \(K\)-theory of global fields |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |