The BC-system and L-functions: geometric counterpart of the adele class space, canonical constructions at finite primes, p-adic KMS theory, and characteristic one

In a 2011 paper built on the Bost-Connes quantum statistical mechanical system, Alain Connes posed five interlocked open problems: give the adele class space of the rational numbers a genuinely geometric, canonical arithmetic structure. The facets are: (1) construct a geometric counterpart $X$ of the adele class space $\mathbb H_{\mathbb Q}=\mathbb A_{\mathbb Q}/\mathbb Q^\times$ whose points are concretely realized as valuations and in which the Galois ambiguity is completely respected, without artificial base-point choices, so that the relation between $X$ and $\mathbb H_{\mathbb Q}$ mirrors the two sides of the class field theory isomorphism; (2) glue the fibers $Y_p$ (mapping tori of Frobenius on embeddings of cyclotomic fields into $\mathbb C_p$) across different primes, which requires a simultaneous uniformization of the embeddings of $\mathbb Q^{\mathrm{cyc}}$ in $\mathbb C_p$; (3) supply the thereby-needed canonical construction of $\overline{\mathbb F}_p$; (4) complete the $p$-adic KMS theory on the open unit disk $M=D(1,1^-)\subset\mathbb C_p$ by classifying the $\mathbb C_p$-valued equilibrium functionals of the integral BC-system (an analytic family of such functionals on $M$ being already constructed), in analogy with the complex case; and (5) construct the characteristic-one objects $R^{\mathrm{un}}$ and the elusive extension (128) of the source relating the sought algebraic closure of the field with one element to $\mathbb Q[\mathbb Q/\mathbb Z]$. Partial progress exists: the zeta sector of the adele class space is realized as points of the scaling site, an analytic family of $p$-adic KMS functionals has been constructed, and a Riemann-Roch theorem over the sphere spectrum is available; the 2026 program papers add a class-field-theoretic functor on the zeta sector, zeta spectral triples, and a Jacobian-type interpretation of the full adele class space as a monoidal extension of a Picard group; but by the authors' own recent statements, a cohomology theory in characteristic one is still missing, and none of the five requested objects or classifications exists.

Background

The Bost-Connes (BC) system, introduced by Bost and Connes in 1995, is a quantum statistical mechanical system attached to the rational numbers. In this setting a system means a C*-dynamical system: a C*-algebra $A$ (a norm-closed, self-adjoint algebra of operators on a Hilbert space, used as the algebra of observables) together with a one-parameter group $t\mapsto\sigma_t$ of automorphisms of $A$ playing the role of time evolution. Its equilibrium states at inverse temperature $\beta>0$ are the KMS (Kubo-Martin-Schwinger) states: states $\varphi$ on $A$ such that for all $a,b\in A$ the function $t\mapsto\varphi(\sigma_t(a)b)$ extends to a function holomorphic in the strip $0<\mathrm{Im}(z)<\beta$ whose upper boundary value is $\varphi(\sigma_{i\beta}(a)b)$. The BC-system has partition function (statistical sum over energy levels) equal to the Riemann zeta function, undergoes a phase transition with spontaneous symmetry breaking at $\beta=1$, and its extremal zero-temperature ($\beta\to\infty$) KMS states are indexed by the embeddings of $\mathbb Q^{\mathrm{cyc}}$ into $\mathbb C$ (equivalently, by the elements of $\hat{\mathbb Z}^\times=\mathrm{Gal}(\mathbb Q^{\mathrm{cyc}}/\mathbb Q)$) and are permuted by the Galois group of the maximal cyclotomic extension $\mathbb Q^{\mathrm{cyc}}$ of $\mathbb Q$ (obtained by adjoining all roots of unity): the zero-temperature phase of the system is the class field theory of $\mathbb Q$, realized by statistical mechanics. Connes's trace-formula work shows that the natural stage for the BC-system and for the spectral side of the zeta function is the adele class space $\mathbb H_{\mathbb Q}=\mathbb A_{\mathbb Q}/\mathbb Q^\times$: the quotient of the adele ring $\mathbb A_{\mathbb Q}$ of $\mathbb Q$ (the restricted product of all completions $\mathbb Q_v$, over the finite primes $p$ and the archimedean place) by the diagonal action of the multiplicative group of nonzero rational numbers. The idele class group $C_{\mathbb Q}=\mathbb A_{\mathbb Q}^\times/\mathbb Q^\times$ (units of the adele ring, again modulo rationals) acts on $\mathbb H_{\mathbb Q}$, and a suitably regularized Lefschetz trace formula (a fixed-point formula for group actions, in the noncommutative-geometric form introduced by Connes) exhibits the nontrivial zeros of the Riemann zeta function as the spectrum of this action. The space $\mathbb H_{\mathbb Q}$ is noncommutative and carries the structure of a hyperring (an algebraic structure like a ring except that addition is allowed to be multivalued); for the function field of a curve over a finite field, the groupoid of prime elements of the corresponding hyperring is canonically the loop groupoid of the maximal abelian cover of the curve. For $\mathbb Q$ no such curve is known, and finding its analogue is the heart of the source problem. The source paper studies what survives of this picture at a fixed prime $p$. Let $\mathbb Q^{\mathrm{cyc}}_p$ be the extension of $\mathbb Q$ generated by roots of unity of order prime to $p$, embedded, after a non-canonical choice, in $\mathbb C_p$, the completion of an algebraic closure of the $p$-adic field $\mathbb Q_p$. The integral form $\mathcal H_{\mathbb Z}$ of the BC-system algebra then admits $\mathbb C_p$-valued functionals satisfying the analogue of the KMS$_\beta$ condition, governed by $p$-adic L-functions (the $p$-adic analogues of Dirichlet $L$-functions, constructed through the Iwasawa logarithm $\log_p$, the $p$-adic replacement of the ordinary logarithm). The natural parameter space for the resulting modular automorphisms is the open unit disk $M=D(1,1^-)\subset\mathbb C_p$, viewed as a multiplicative group, via $\beta=\log_p(\sigma)/\log_p(1+q)$, where $q=p$ (with $q=4$ when $p=2$) is the base of the $p$-adic normalization used to define the disk. The paper also associates to each prime $p$ the space $Y_p$, the mapping torus (i.e. the quotient $Y_p=(X_p\times(0,1))/\mathord{\cong}$ with $(\phi,t)\cong(\phi\circ\mathrm{Fr},t^p)$) of the action of the Frobenius automorphism (the map $x\mapsto x^p$ on finite-field points) on the set $X_p$ of embeddings of $\mathbb Q^{\mathrm{cyc}}_p$ into $\mathbb C_p$. The source shows that the quotient $\Sigma_p=X_p/\mathrm{Gal}(\overline{\mathbb F}_p/\mathbb F_p)$ (the Galois group acting by composition) is 'not what one wants as a fiber over $p$' and that the finer space $Y_p$ is the fine fiber over $p$ of the sought geometric space: the total space of a principal bundle over $\Sigma_p$ with structure group a connected compact solenoid (an inverse limit of circles). Comparing with $\mathbb H_{\mathbb Q}$ shows that $Y_p$ embeds in a quotient of $E(\mathbb C_p)\times(0,1)$, where $E(A)=\mathrm{Hom}(\mathbb Z[\mathbb Q/\mathbb Z],A)$ is the set of homomorphisms from the group ring of the torsion group $\mathbb Q/\mathbb Z$ into $A$. Matching these spaces for different primes forces a simultaneous, canonical uniformization of the embeddings $\mathbb Q^{\mathrm{cyc}}\hookrightarrow\mathbb C_p$, and it is at this point, in the source's words, that 'appears the need for a canonical construction of $\overline{\mathbb F}_p$': an algebraic closure of the finite field with $p$ elements built without arbitrary choices, of the kind studied in Lenstra's work on finding isomorphisms between finite fields. The final facet belongs to characteristic one. A semifield is a field-like structure lacking additive inverses; the basic example is $\mathbb R^{\max}_+=\mathbb R\cup\{-\infty\}$ with operations $\max$ and $+$ (the max-plus or tropical semifield). The source paper proves that the Witt construction (the classical device producing, from a ring, the ring of big Witt vectors that underlies $p$-adic number systems and $p$-adic L-functions) extends to characteristic one, and it searches for an object $R^{\mathrm{un}}$ playing the role of a maximal unramified extension in this setting; a certain field $E$ is exhibited as a first hint toward $R^{\mathrm{un}}$, but the expected extension relating the still-unknown algebraic closure of the field with one element $\mathbb F_1$ (the hypothetical base over which the integers would be a polynomial ring, as finite fields are for function fields) to $\mathbb Q[\mathbb Q/\mathbb Z]$ — labeled (128) in the source — 'remains elusive'. Subsequent work of Connes and Consani developed this program: the arithmetic site (a topos, i.e. a category-theoretic generalization of a space, dual to the profinite integers, equipped with a structure sheaf of tropical semirings, whose points over $\mathbb R^{\max}_+$ recover the quotient of the adele class space by the maximal compact subgroup $\hat{\mathbb Z}^\times$, that is its zeta sector); the scaling site (a topos built from the half-line $[0,\infty)$ under the action of the positive rationals, with a structure sheaf of convex functions), whose points over $\mathbb R^{\max}_+$ realize the zeta sector; a real-valued Riemann-Roch theorem (Riemann-Roch being the classical formula linking the degree of a divisor to the dimensions of the spaces of functions with controlled poles); the universal thickening of $\mathbb R$ (an archimedean analogue of the rings of $p$-adic periods); and an algebraic geometry over the sphere spectrum $S$ of stable homotopy theory, whose rings are Segal's gamma rings (functorial counterparts of rings introduced by Graeme Segal), with $\mathbb Z[\mathbb Q/\mathbb Z]$ identified with the degree-zero part of the Witt ring of a monoid-algebra 'algebraic closure' of $S$. By the authors' own recent assessment, however, these constructions realize only the zeta sector, and the ingredients needed to extend the Weil-style positivity proof of the Riemann hypothesis for function fields to $\mathbb Q$ — a Riemann-Roch formula on the square of the scaling site and a cohomology theory in characteristic one beyond $H^0$ — are still missing. The 2026 program papers continue this line without closing it: 'Knots, primes and class field theory' constructs a functor from the finite abelian extensions of $\mathbb Q$ to the finite covers of the zeta sector $X_{\mathbb Q}$ of the adele class space (the sector corresponding to the Riemann zeta function), with the Frobenius at $p$ realized as the monodromy of the periodic orbits of length $\log p$ under the scaling action; 'Zeta spectral triples' (with Henri Moscovici) conditions spectral triples so that their low-lying spectra resemble the first zeta zeros; and 'On the Jacobian of $\overline{\mathrm{Spec}\,\mathbb{Z}}$' interprets the full adele class space of $\mathbb Q$ as a natural monoidal extension of the Picard group (the group of line bundles) of the arithmetic curve $\overline{\mathrm{Spec}\,\mathbb{Z}}$ (the scheme obtained by adjoining to the integers all roots of unity), identifying its elements with torsion-free rank-one abelian groups endowed with rigidifying data. Each of these works interprets or acts on the adele class space or on a sector of it; none constructs the choice-free geometric counterpart $X$ demanded by the source.

Problem Statement

The problem, as posed in the source paper (Alain Connes, 'The BC-system and L-functions', Japanese Journal of Mathematics 6 (2011)), is one interlocked objective: to give the adele class space of the rational numbers a genuinely geometric and arithmetic structure that is canonical at every place and in which the equilibrium states of the BC-system and the spectral realization of the zeta zeros live without artificial choices. Its five linked facets are: (1) Construct a geometric counterpart $X$ of the adele class space $\mathbb H_{\mathbb Q}=\mathbb A_{\mathbb Q}/\mathbb Q^\times$ (endowed with the natural action of the idele class group $C_{\mathbb Q}$) in which the points $v\in X$ are concretely realized as valuations and the Galois ambiguity is completely respected, without making an artificial choice of a base point over each place; the relation between $X$ and the adele class space should be of the same nature as the relation between the two sides of the class field theory isomorphism. (2) Perform the gluing of the fibers $Y_p$ (the mapping tori of the Frobenius action on the embedding spaces $X_p=\mathrm{Hom}(\mathbb Q^{\mathrm{cyc}}_p,\mathbb C_p)$) for different primes, since all these fibers must embed in the same noncommutative space to account for the transversality factors in the explicit formulas; this requires uniformizing the embeddings of $\mathbb Q^{\mathrm{cyc}}$ in $\mathbb C_p$ for the different primes simultaneously. (3) Provide the canonical construction of $\overline{\mathbb F}_p$ (a choice-free algebraic closure of the finite field with $p$ elements) that is needed at this point, so that the spaces $X_p$, the Galois-theoretic fibers $\Sigma_p$, and the mapping tori $Y_p$ fit together. (4) Complete the $p$-adic KMS theory on the covering $M=D(1,1^-)\subset\mathbb C_p$ (the open unit disk viewed as a multiplicative group): the construction of an analytic family of $\mathbb C_p$-valued KMS$_\beta$ functionals on $M$ being known (reported in the source from its companion paper), prove the classification of these functionals and their uniqueness properties, in exact analogy with the classification of KMS states in the complex case. (5) In characteristic one: construct the analogue $R^{\mathrm{un}}$ of a maximal unramified extension, for which the source's field $E$ is a first hint, and elucidate the unknown extension (128) relating the sought algebraic closure of the field with one element to $\mathbb Q[\mathbb Q/\mathbb Z]$, which the source describes as remaining elusive and as possibly involving the dual system of the BC-system. The five facets form a single interlocked objective, as the source presents them, to be realized as compatible parts of a single structure; the partial resolutions of individual facets (realization of the zeta sector, construction without classification of the $p$-adic KMS functionals, cohomological progress over the sphere spectrum, and the 2026 Jacobian-type moduli interpretation of the adele class space) leave that interlocked problem open.

References

1. Alain Connes, 'The BC-system and L-functions', Japanese Journal of Mathematics 6 (2011) 1-44, DOI 10.1007/s11537-011-1035-0, https://doi.org/10.1007/s11537-011-1035-0 (source paper; full text at https://alainconnes.org/wp-content/uploads/TheBC-systemAndL-functions.pdf)
2. J.-B. Bost and A. Connes, 'Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory', Selecta Mathematica 1 (1995) 411-457, DOI 10.1007/BF01589495, https://doi.org/10.1007/BF01589495
3. Alain Connes, 'Trace formula in noncommutative geometry and the zeros of the Riemann zeta function', Selecta Mathematica 5 (1999) 29-106, DOI 10.1007/s000290050042, https://doi.org/10.1007/s000290050042
4. Alain Connes and Caterina Consani, 'On the arithmetic of the BC-system', Journal of Noncommutative Geometry 8 (2014) 873-945, DOI 10.4171/JNCG/173, arXiv:1103.4672, https://doi.org/10.4171/JNCG/173
5. H. W. Lenstra, 'Finding isomorphisms between finite fields', Mathematics of Computation 56 (1991) 329-347, DOI 10.1090/S0025-5718-1991-1052099-2, https://doi.org/10.1090/S0025-5718-1991-1052099-2
6. Bart de Smit and Hendrik Lenstra, 'Standard models for finite fields', in Handbook of Finite Fields, CRC Press, 2013, pp. 401-404, DBLP record books/crc/13/SmitL13, https://dblp.org/rec/books/crc/13/SmitL13.html
7. Alain Connes and Caterina Consani, 'Geometry of the arithmetic site', Advances in Mathematics 291 (2016) 274-329, DOI 10.1016/j.aim.2015.11.045, arXiv:1502.05580, https://doi.org/10.1016/j.aim.2015.11.045
8. Alain Connes and Caterina Consani, 'Geometry of the scaling site', Selecta Mathematica 23 (2017) 1803-1850, DOI 10.1007/s00029-017-0313-y, https://doi.org/10.1007/s00029-017-0313-y
9. Alain Connes and Caterina Consani, 'Universal thickening of the field of real numbers', in Advances in the Theory of Numbers, Fields Institute Communications 77, 2015, pp. 11-74, DOI 10.1007/978-1-4939-3201-6_2, https://doi.org/10.1007/978-1-4939-3201-6_2
10. Alain Connes, 'An essay on the Riemann hypothesis', in Open Problems in Mathematics, Springer, 2016, pp. 225-257, DOI 10.1007/978-3-319-32162-2_5, arXiv:1509.05576, https://doi.org/10.1007/978-3-319-32162-2_5
11. Alain Connes and Caterina Consani, 'Segal's gamma rings and universal arithmetic', The Quarterly Journal of Mathematics 72 (2021) 1-29, DOI 10.1093/qmath/haaa042, https://doi.org/10.1093/qmath/haaa042
12. Alain Connes and Caterina Consani, 'BC-system, absolute cyclotomy and the quantized calculus', EMS Surveys in Mathematical Sciences 9 (2023) 447-475, DOI 10.4171/EMSS/64, arXiv:2112.08820, https://doi.org/10.4171/EMSS/64
13. Alain Connes and Caterina Consani, 'Riemann–Roch for the ring ℤ', Comptes Rendus. Mathematique 362 (2024) 229-235, DOI 10.5802/crmath.543, https://doi.org/10.5802/crmath.543
14. Bora Yalkinoglu, 'Bost-Connes systems and periodic Witt vectors', preprint, 2025, arXiv:2507.23759, https://arxiv.org/abs/2507.23759
15. Alain Connes and Caterina Consani, 'Knots, primes and class field theory', in Regulators V, Contemporary Mathematics 842, American Mathematical Society, 2026, pp. 105-132, DOI 10.1090/conm/842/16852, arXiv:2501.06560, https://doi.org/10.1090/conm/842/16852
16. Alain Connes, Caterina Consani and Henri Moscovici, 'Zeta spectral triples', in Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time, EMS Series of Lectures in Mathematics 37, European Mathematical Society, 2026, pp. 39-76, DOI 10.4171/elm/37/3, arXiv:2511.22755, https://doi.org/10.4171/elm/37/3
17. Alain Connes and Caterina Consani, 'On the Jacobian of $\overline{\mathrm{Spec}\,\mathbb{Z}}$', preprint, 2026, arXiv:2602.15941, https://arxiv.org/abs/2602.15941