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Deligne periods of mixed motives, \(K\)-theory and the entropy of certain \(\mathbb Z^n\)-actions. (English) Zbl 0913.11027
The underlying paper relates aspects of the theory of dynamical systems to mixed motives. The link is suggested by integrals of the form \(\int\log| f| {dz\over z}\) which occur in older work of Beilinson on regulators relating algebraic \(K\)-theory to what is called Deligne-Beilinson cohomology (of analytic varieties). For \(f=P\) a Laurent polynomial, it turns out that the above integrals can be interpreted as the entropy \(h(\mathcal{O}_Z)\) of a \({\mathbb Z}^n\)-action on the Pontryagin dual of the space of global sections of the structure sheaf \(\mathcal{O}_Z\) of an irreducible closed subscheme \(Z\) of the split \(n\)-torus \({\mathbb G}^n_{m,{\mathbb Z}}\). This entropy (more precisely, its exponential) can then be identified with what is called the Mahler measure \(m(P)\) (more precisely, \(M(P)=\exp m(P)\)) of the polynomial. In formula: for any \(P\in\Gamma({\mathbb G}^n_{m,{\mathbb Z}},\mathcal{O})={\mathbb Z}[{\mathbb Z}^n]={\mathbb Z}[t_1^{\pm 1},\ldots,t_n^{\pm 1}]\), \(P\neq 0\), and \(Z=\text{Spec}({\mathbb Z}[{\mathbb Z}^n]/(P))\), one has \[ h(\mathcal{O}_Z)=m(P):={1\over(2\pi i)^n}\int_{T^n}\log| P(z_1,\ldots,z_n)| {dz_1\over z_1}\cdots{dz_n\over z_n}, \] where \(T^n=S^1\times\cdots\times S^1\) is the real \(n\)-torus. In Beilinson’s world, the associated \((n+1)\)-fold cup product \(\log| P| \cup\log| z_1| \cup\cdots\cup\log| z_n| \) is just the image \(r_{\mathcal D}(\{P,t_1,\ldots,t_n\})\) of the regulator map of the \(K\)-theory symbol \(\{P,t_1,\ldots,t_n\}\) in Deligne-Beilinson cohomology of \({\mathbb G}^n_{m,{\mathbb Z}}\setminus Z\). It is known that, according to ideas of Deligne, algebraic \(K\)-theory should be related to mixed motives. In this respect, one may associate a mixed motive to the symbol \(\{P,t_1,\ldots,t_n\}\) and it is shown that the Deligne period of this mixed motive is just \(m(P)\).
Take a non-zero Laurent polynomial \(P\in{\mathbb Q}[{\mathbb Z}^n]\) and let \(X_P/{\mathbb Q}\) be the complement of the zero locus of \(P\) in \({\mathbb G}_{m,{\mathbb Q}}^n\). \(X_P\) may be considered as a closed subvariety of \({\mathbb G}^{n+1}_{m,{\mathbb Q}}\) via the embedding of the graph of \(P\). Now on \({\mathbb G}^{n+1}_m\) one has the action of \(\Gamma_{n+1}:=\mu_2^{n+1}\rtimes\mathfrak S_{n+1}\). One writes \(\varepsilon\) for the character which is the product on \(\mu_2^{n+1}\) and the sign character of \(\mathfrak S_{n+1}\). For any subgroup \(\Gamma\subset\Gamma_{n+1}\) one may consider \[ X=\coprod_{\gamma\in\Gamma}X_P^{\gamma}\rightarrow{\mathbb G}_m^{n+1},\quad\text{resp.} \quad X=\bigcup_{\gamma\in\Gamma}X_P^{\gamma}\rightarrow{\mathbb G}_m^{n+1}, \] the sum, resp. the union, of the \(\Gamma\)-translates of \(X_P\). In both cases, \(X\) is an affine \(n\)-dimensional closed subvariety of \({\mathbb G}_m^{n+1}\). Writing \(H^{\bullet}(*)(\varepsilon)\) for the \(\varepsilon\)-isotypical component of \(H^{\bullet}(*)\), one constructs an extension \[ 0\rightarrow H^n(X,n+1)(\varepsilon)\rightarrow N\rightarrow{\mathbb Q}(0)\rightarrow 0 \] and its twisted dual \[ 0\rightarrow{\mathbb Q}(1)\rightarrow M=N^{\vee}(1)\rightarrow H_n(X,-n)(\varepsilon)\rightarrow 0. \] Let \(\omega_{\mathcal H}\in F^0N_{dR}\) be the canonical class corresponding to \(1\in{\mathbb Q}\) under \(F^0N_{dR}{\buildrel\sim\over\longrightarrow}F^0{\mathbb Q}(0)_{dR}={\mathbb Q}\). For the duals one has \(M^+_B\simeq H^B_n(X,{\mathbb Q}(-n))^+(\varepsilon)\). One has the period pairing \(\langle , \rangle:M^+_B\times F^0N_{dR}\rightarrow{\mathbb R}\). Let \(i:X_P\hookrightarrow X\) be the inclusion, and write \(i_*[T^n](\varepsilon)\) for the \(\varepsilon\)-isotypical component of \(i_*[T^n]\) in \(H_n(X({\mathbb C}),{\mathbb Q})\). Finally, let \(c\in M^+_B\simeq H_n^B(X,{\mathbb Q}(-n))^+(\varepsilon)\) correspond to the cycle \(i_*[T^n](\varepsilon)\). One has the result: Let \(P\in{\mathbb Q}[t_1^{\pm 1},\ldots,t_n^{\pm 1}]\) be without zeroes on \(T^n\). Then, under the period pairing, \(\langle c,\omega_{\mathcal H}\rangle = m(P)\). If \(P\in{\mathbb Z}[t_1^{\pm 1},\ldots,t_n^{\pm 1}]\) one gets an interpretation of \(\langle c,\omega_{\mathcal H}\rangle\) as the entropy of a natural \({\mathbb Z}^n\)-action. In A. Huber’s category \(\mathcal{MM}\) of mixed motives one has the Chern character \[ \text{ch}:H^{n+1}_{\mathcal M}(X_P,{\mathbb Q}(n+1))\rightarrow\text{Ext}^1_{\mathcal{MM}}({\mathbb Q}(0),H^n(X_P,n+1))=\text{Ext}^1_{\mathcal{MM}}({\mathbb Q}(0),H^n(X,n+1)(\varepsilon)) \] and one shows that \(\text{ch}(\{P,t_1,\ldots,t_n\})\) gives the extension \(N\).
One can also consider symbols and their relation to differences of Mahler measures of the form \(m(P^*)-m(P)\), where \(P^*(t_1,\ldots,t_{n-1})=a_{i_0}(t_1,\ldots,t_{n-1})\) if \[ P(t_1,\ldots,t_n)=\sum_{i\geq 0}a_i(t_1,\ldots,t_{n-1})t_n^i\in{\mathbb C}[t_1^{\pm 1},\ldots,t_n^{\pm 1}]. \] It turns out that such differences can also be expressed as evaluations of \(n\)-fold cup products in Deligne cohomology against suitable homology classes. This can also be given a motivic description. A particularly interesting example is the case \(P(t_1,t_2)=t_1^2t_2+t_1t_2^2+t_1t_2+t_1+t_2\), thus \(P^*(t_1)=P(t_1,0)=t_1\). The projective completion of the related variety \(Z\) turns out to be the elliptic curve \(E:y^2+xy+y=x^3+x^2\), which is isogenous to \(X_0(15)\). One has \(m(P^*)=0\) and \(m(P)\) becomes (up to a minus sign) the evaluation of \(r_{\mathcal D}([t_1,t_2])\) on an element of \(H_1(E/{\mathbb R},{\mathbb Q}(-1))\) where \([t_1,t_2]\in H^2_{\mathcal M}(E,{\mathbb Q}(2))\). This leads to an expression of \(m(P)\) in terms of an Eisenstein-Kronecker-Lerch series. This is closely related to \(L(E,2)\) modulo \({\mathbb Q}^{\times}\) according to the Beilinson conjectures.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
19E08 \(K\)-theory of schemes
28D20 Entropy and other invariants
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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