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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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