The BC-system and \(L\)-functions
This article is based on the 8th Takagi Lectures delivered by the author at the Research Institute for Mathematical Sciences of Kyoto University on November 2010. The paper surveys on some relations between \(L\)-functions and the Bost-Connes (BC-) system in noncommutative arithmetic geometry ([\textit{J.-B. Bost} and \textit{A. Connes}, Sel. Math., New Ser. 1, No. 3, 411--457 (1995; Zbl 0842.46040); \textit{A. Connes}, Sel. Math., New Ser. 5, No.1, 29--106 (1999; Zbl 0945.11015)]) also reporting on recent results obtained in collaboration with the reviewer [\textit{A.~Connes} and \textit{C.~Consani}, Compos. Math. 146, No. 6, 1383--1415 (2010; Zbl 1201.14001); \textit{A.~Connes} and \textit{C.~Consani}, in: G. van Dijk (ed.) et al., Casimir force, Casimir operators and Riemann hypothesis. Mathematics for innovation in industry and science. Proceedings of the conference, Fukuoka, Japan, November 9--13, 2009. Berlin: de Gruyter. 147--198 (2010; Zbl 1234.14002); \textit{A.~Connes} and \textit{C.~Consani}, J. Number Theory 131, No. 2, 159--194 (2011; Zbl 1221.14002); \textit{A.~Connes} and \textit{C.~Consani}, ``On the arithmetic of the BC-system'', \url{arXiv:1103.4672}, to appear in J. Noncommut. Geom.; \textit{A.~Connes} and \textit{C.~Consani}, ``Characteristic one, entropy and the absolute point'', in: Noncommutative Geometry, Arithmetic, and Related Topics, the Twenty-First Meeting of the Japan-U.S. Mathematics Institute, Baltimore 2009, JHUP. 75--139 (2012); \url{arXiv:0911.3537}] on the quest for an ``absolute arithmetic'' and on new relations of the BC-system with \(p\)-adic analysis and the theory of Witt vectors. The link between class field theory and type III factors in operators algebra that was discovered by the author several years ago is established by a quantum statistical mechanical system (the BC-system) that connects the theory of factors and the KMS condition in quantum mechanics through the implementation of a noncommutative space of class field theory origin: the adèle class space. In \S 2 it is recalled that the noncommutative (integral) Hecke algebra \({\mathcal H}_\mathbb Z\) that describes the BC-system admits a presentation (with generators \(x\in \mathbb Z[\mathbb Q/\mathbb Z]\), \(\mu_n^*\) and \(\tilde\mu_n\), \(n\in\mathbb N\)) as the crossed product \(\mathbb Z[\mathbb Q/\mathbb Z]\rtimes_{\tilde\rho}\mathbb N\) of the group ring \(\mathbb Z[\mathbb Q/\mathbb Z]\) by ring endomorphisms \(\sigma_n\) with associated quasi-inverse additive maps \(\tilde\rho_n\) fulfilling fundamental commutation relations. The complexified algebra \({\mathcal H}_\mathbb C\) already has a highly non-trivial structure since its regular representation, in the Hilbert space of one sided classes, generates a factor of type III\(_1\) and a canonical ``time evolution'' \(\sigma_t\in \Aut({\mathcal H}_\mathbb C)\). It is the study of the KMS-equilibrium states at different temperatures that reveals the arithmetic nature of this dynamical system in view of the following facts: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] The partition function of the system is the Riemann zeta function. \item[{\(\bullet\)}] There is a phase transition with spontaneous symmetry breaking at the pole of the Riemann zeta function. \item[{\(\bullet\)}] The zero temperature vacuum states implement the global class field isomorphism for \(\mathbb Q\). \end{itemize}} In \S 3, the author recalls that this interplay between thermodynamics and number theory has been recently extended at a non-Archimedean place [\url{arXiv:1103.4672}]. For each rational prime \(p<\infty\), there is a strong relation connecting the integral BC-system and the universal Witt ring \(\mathbb W_0(\bar\mathbb F_p)\) of an algebraic closure of a prime field. The Witt construction is viewed as a K-theory endofunctor \(A \mapsto \mathbb W_0(A) = K_0(\mathrm{End} A)/K_0(A)\), in the category of commutative rings (with unit). Let \(X_p\) be the space of all injective group homomorphisms \(\sigma:\bar\mathbb F_p^\times \to \mathbb Q/\mathbb Z\) and denote by \((\mathbb Q/\mathbb Z)^{(p)}\) their common range. Then the following result holds (cf. Theorem~3.2) Theorem. To any \(\sigma\in X_p\) corresponds a ring isomorphism \[ \tilde\sigma: \mathbb W_0(\bar \mathbb F_p)\overset{\sim}{\longrightarrow}\mathbb Z[(\mathbb Q/\mathbb Z)^{(p)}]\subset \mathbb Z[\mathbb Q/\mathbb Z]. \] Moreover the Frobenius \(F_n\) and Verschiebung maps \(V_n\) on \(\mathbb W_0(\bar \mathbb F_p)\) are obtained by restriction of the ring endomorphisms \(\sigma_n\) and the maps \(\tilde\rho_n\) on \(\mathbb Z[\mathbb Q/\mathbb Z]\) by the formulas \[ \tilde\sigma\circ F_n=\sigma_n\circ \tilde\sigma\,, \;\;\tilde\sigma\circ V_n=r\circ\tilde\rho_n\circ \tilde\sigma \] (\(r: \mathbb Z[\mathbb Q/\mathbb Z]\to \mathbb Z[(\mathbb Q/\mathbb Z)^{(p)}]\) is the natural retraction). Thus the \(p\)-primary structure of the integral BC-system is completely encoded by the universal ring \(\mathbb W_0(\bar\mathbb F_p)\), with a precise dictionary expressing the key operators \(\sigma_n\) and \(\tilde\rho_n\) as respectively the Frobenius \(F_n\) and Verschiebung \(V_n\) on \(\mathbb W_0(\bar\mathbb F_p)\). For each embedding \(\sigma\) of the multiplicative group of an algebraic closure of \(\mathbb F_p\) as complex roots of unity, one then constructs a \(p\)-adic indecomposable representation \(\pi_\sigma\) of the integral BC-system as additive endomorphisms of the big Witt ring of \(\bar\mathbb F_p\) (cf. Theorems~3.3 and 3.6) Theorem. Let \(\sigma\in X_p\). The following formulas define a representation \(\pi_\sigma\) of the integral BC-algebra \(\mathcal H_\mathbb Z\) on \(\mathbb W(\bar \mathbb F_p)\). For \(n\in I(p)\) and for \(x\in \mathbb Z[\mathbb Q/\mathbb Z]\), \(\pi_\sigma (\mu_n)\), \(\pi_\sigma(x)\) and \(\pi_\sigma(\mu_n^*)\) are \(\mathcal O\)-linear operators on \(\mathbb W(\bar\mathbb F_p)\) \[ \begin{aligned} \pi_\sigma (\mu_n)\epsilon_m &= \epsilon_{nm}, \\ \pi_\sigma(e(a/b))\epsilon_m&=\rho(\zeta_{a/b}^m)\epsilon_m ,\quad\forall a \in \mathbb Z,\;\forall b, m\in I(p)\\ \pi_\sigma(\mu_n^*)\epsilon_k&=\begin{cases} 0 & \text{if}\, k\notin n\mathbb N\, \\ \epsilon_{k/n} & \text{if}\, k\in n\mathbb N\,. \end{cases} \end{aligned} \] One has \(\pi_\sigma(x)=\pi_\sigma(r(x))\) for all \(x\in \mathbb Z[\mathbb Q/\mathbb Z]\) and \[ \pi_\sigma (\mu_p)=\mathrm{Fr}^{-1},\quad \pi_\sigma(\mu_p^*) = \mathrm{Fr}. \] Here \(\mathrm{Fr}\) denotes the Frobenius automorphism, acting componentwise as a skew-linear operator and \(I(p)\subset\mathbb N\) is the set of integers prime to \(p\). The obtained representations \(\pi_\sigma\) are the \(p\)-adic analogues of the complex, extremal KMS\(_\infty\) states of the BC-system \(\mathcal H_\mathbb C\) (cf.[Zbl 0842.46040]). In \S 3.6 the author recalls (cf. [\url{arXiv:1103.4672}]) that the role of the Riemann zeta function, as partition function of the BC-system over \(\mathbb C\) is replaced, in the \(p\)-adic case, by the \(p\)-adic \(L\)-functions and the polylogarithms whose values at roots of unity encode the KMS states. Let \(M=D(1,1^-)\) be the open unit disk in \(\mathbb C_p\) with radius \(1\), viewed as a multiplicative group covering of \(\mathbb C_p\). Then the following result holds Theorem. There exists an analytic family of functionals \(\psi_{\lambda,\rho}\), \(\lambda\in M\), on \(\mathcal H_\mathbb Z^{(p)}\) such that {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] \(\psi_{\lambda,\rho}(1)=1.\) \item[{\(\bullet\)}] \(\psi_{\lambda,\rho}\) fulfills the KMS condition \[ \psi_{\lambda,\rho}(x\sigma[\lambda](y))=\psi_{\lambda,\rho}(y\,x),\quad\forall x,y\in \mathcal H_{\mathbb C_p}^{(p)}. \] \item[{\(\bullet\)}] For \(\beta\in D_p\) and \(\lambda=(1+q)^\beta\) one has \[ \psi_{\lambda,\rho}=Z(\beta)^{-1}\varphi_{\beta,\rho} \] where \(Z(\beta):= \varphi_{\beta,\rho}(1)=L_p(\beta,1)\) denotes the \(p\)-adic \(L\)-function. \end{itemize}} This result shows in particular that the division relations for the \(p\)-adic polylogarithms at roots of unity correspond to the KMS condition. Iwasawa theory is implemented to extend the KMS theory to the covering \(M\) of \(\mathbb C_p\). In \S\S 4.3,4 the author reports on his recent work (cf. [Zbl 1221.14002]) on the hyperring structure of the adèle class space that combines with \(p\)-adic analysis to refine the space \(\mathrm{Val}_p(\mathbb Q^{\mathrm{cyc}})\) of valuations of the cyclotomic extension of \(\mathbb Q\) in terms of a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the sought for ``arithmetic site''. To accomplish this result a description of the BC-system has been developed (cf. [\url{arXiv:1103.4672}]) as an affine pro-group scheme \(\mathcal E\) over \(\mathbb Z\) enriched with the dynamic of the action of a semigroup of endomorphisms. This description allows one to consider the rational points \(\mathcal E(A) = \Hom(\mathbb Z[\mathbb Q/\mathbb Z],A)\) of \(\mathcal E\) over any ring \(A\) and to implement, for each rational prime \(p\) (inclusive of \(\infty\)), a canonical inclusion \[ X_p\subset \Hom(\mathbb Z[\mathbb Q/\mathbb Z],\mathbb C_p) = \mathcal E(\mathbb C_p).\tag{1} \] The quotient of \(X_p\) by \(\Aut(\bar \mathbb F_p) = f_p^{\hat\mathbb Z}\) (\(f_p=\) Frobenius) is isomorphic to \(\mathrm{Val}_p(\mathbb Q^{\mathrm{cyc}})\). In \S 4.4, the author reviews and comments on the result that the space \[ X(\mathbb C_p):= \left( \mathcal E(\mathbb C_p)\times (0,\infty)\right)/(\mathbb N\times \{\pm 1\}) \] matches, for any \(p\) including \(\infty\), the definition of the adèle class space \(\mathbb H_\mathbb Q = \mathbb A_\mathbb Q/\mathbb Q^\times\) as a hyperring (cf. [Zbl 1221.14002]). The action of \(m=\pm n\) (in the semigroup \(\mathbb N\times \{\pm 1\}\)) is the product of the linearization of the action \(e(\gamma)\mapsto e(m\gamma)\) on the (\(\mathbb C_p\)-rational points of the) scheme \(\mathcal E\), with the action on \((0,\infty)\) given by the map \(x\mapsto x^m\). One derives the following Theorem. {\parindent=6mm \begin{itemize}\item[1.] The space \(X(\mathbb C)\) is canonically isomorphic to the adèle class space \(\mathbb H_\mathbb Q\). \item[2.] The subspace of the adèle class space made by classes whose Archimedean component vanishes corresponds to the quotient \[ \mathcal E(\mathbb C)/(\mathbb N\times \{\pm 1\})=\hat\mathbb Z/(\mathbb N\times \{\pm 1\}). \] \end{itemize}} Note that by using the inclusion \((0,1)\subset (0,\infty)\), one derives a natural inclusion \[ Y_p=\left(X_p\times (0,1)\right)/\sigma^\mathbb Z\to \left( \mathcal E(\mathbb C_p)\times (0,\infty)\right)/(\mathbb N\times \{\pm 1\})=X(\mathbb C_p). \] For \(p=\infty\) one has the natural inclusion \[ Y_\infty=\Hom(\mathbb Q^{\mathrm{cyc}},\mathbb C)/\{\pm 1\}\to \left( \mathcal E(\mathbb C)\times (0,\infty)\right)/(\mathbb N\times \{\pm 1\})=X(\mathbb C) \] which is obtained by using the canonical inclusion (1) for \(p=\infty\) and the fixed point \(1\in (0,\infty)\). The final output is summarized by the following Theorem. Let \(A\) be a commutative ring. {\parindent=6mm \begin{itemize}\item[1.] The abelian group \(\mathcal E(A)\) is torsion free. \item[2.] The space \[ X(A)=\left(\mathcal E(A)\times (0,\infty)\right)/(\mathbb N\times \{\pm 1\})\tag{2} \] is a module over the hyperring \(\mathbb H_\mathbb Q\). \item[3.] For any rational prime \(p\), \(X(\mathbb C_p)\) is a free module of rank one over \(\mathbb H_\mathbb Q\). \end{itemize}} \S 4.1 overviews on the recent results obtained in [Zbl 1201.14001],[Zbl 1221.14002],[\url{arXiv:0911.3537}] on the quest for an ``absolute arithmetic'' by implementing tools of noncommutative geometry with the goal to supply the definition, in characteristic zero, of a geometric curve supporting the arithmetic of the Riemann zeta function. The author refers to the study of the ``curve'' \(C=\overline {\mathrm{Sp}\,\mathbb Z}\) over \(\mathbb F_1\), whose zeta function \(\zeta_C(s)\) would agree with the complete Riemann zeta function \(\zeta_\mathbb Q(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)\). Inspired by the pioneering work of R. Steinberg and J. Tits, \textit{C.~Soulé} associated in [Mosc. Math. J. 4, No. 1, 217--244 (2004; Zbl 1103.14003)] a zeta function to any sufficiently regular counting-type function \(N(q)\), by considering the following limit \[ \zeta_N(s):=\lim_{q\to 1}Z(q,q^{-s}) (q-1)^{N(1)}\qquad s\in\mathbb R.\tag{3} \] Here, \(Z(q,q^{-s})\) denotes the evaluation, at \(T=q^{-s}\), of the Hasse-Weil zeta function \[ Z(q,T) := \exp\left(\sum_{r\geq 1}N(q^r)\frac{T^r}{r}\right).\tag{4} \] For the consistency of the definition (3), the counting function \(N(q)\) should be defined for all real numbers \(q \geq 1\) and not only for prime integers powers as for the classical counting function in (4). Definition (3) was mainly motivated by the idea to introduce a notion of zeta function over \(\mathbb F_1\) whose description is in agreement with the function predicted by \textit{Yu.~Manin} [Astérisque 228, 121--163 (1995; Zbl 0840.14001)] in the context of a theory of absolute motives. However, definition (3) is too naive to be the correct one. By following the interpretation of \(N(1)\) (cf. (3)) given in [Zbl 1103.14003], this number should coincide with the Euler characteristic of the curve \(C\). On the other hand, since \(C\) should have infinite genus, one deduces that \(N(1)=-\infty\), in apparent contradiction with the expected positivity of \(N(q)\), for \(q>1\). In \S 4.1 the author overviews the results obtained in [Zbl 1201.14001] and [Zbl 1234.14002] where this difficulty has been solved by replacing (3) with an integral formula and by computing explicitly the distribution \(N(q)\) which fulfills the expected positivity, for \(q>1\), and the divergence at \(q=1\). Finally, \S 5 reports on recent results of the author (cf. [\textit{A.~Connes}, Contemp. Math. 546, 83--113 (2011; Zbl 1245.13015)]) on the analogue of the \(p\)-isotypical Witt construction \(\mathbb W_{p^\infty}\) in characteristic one, when this construction is applied to the semifield \(\mathbb R_+^{max}\) of tropical geometry/idempotent analysis. We refer to the recent survey of [loc. cit.] made by this reviewer in Zbl 1245.13015.
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