The theory of endomotives — noncommutative spaces combining Artin motives with semigroup dynamics, introduced by Connes, Consani and Marcolli to place the Bost–Connes quantum statistical mechanical system in a motivic framework — was originally defined only over reduced algebras, in agreement with the classical definition of Artin motives. In 'Fun with F_1' (J. Number Theory 129 (2009) 1532–1561), the authors show that the Bost–Connes endomotive descends to the 'field with one element' and analyze its reduction at primes: over a perfect field of characteristic p the construction produces genuinely unreduced (nilpotent) Artinian algebras, the endomorphisms realize the Frobenius correspondence, and the partial inverses only survive away from p. The paper therefore poses the problem of a general construction of endomotives in arbitrary characteristic: a framework based on inductive systems of augmented commutative Artinian K-algebras with commutative families of correspondences, required to determine homomorphisms such as the rho_n when denominators are allowed and to contain the endomotives of self-maps of pointed varieties including ramified maps, together with a systematic comparison of the two non-equivalent reduction procedures and a formulation of the associated KMS-type thermodynamics in positive characteristic. No unreduced endomotive theory valid in arbitrary characteristic has been constructed to date: subsequent work developed p-adic representations of the BC-system, characteristic-zero functoriality, function-field analogues, categorifications and q-deformations, but not such a theory.

Background

A quantum statistical mechanical system in the sense of the Bost–Connes construction is a C*-algebra (a norm-closed algebra of bounded operators on a Hilbert space, closed under adjoints) equipped with a one-parameter automorphism group describing time evolution. The Bost–Connes (BC) system, built from the Hecke algebra (the convolution algebra of locally constant, compactly supported functions) of the 'ax+b' algebraic group over the rationals, has the Riemann zeta function as partition function, a phase transition with spontaneous symmetry breaking at the pole of zeta, and zero-temperature equilibrium states whose symmetry group realizes the Galois group of the maximal abelian extension of Q — that is, the class field theory of Q appears as a thermodynamic symmetry. Equilibrium states are KMS states (Kubo–Martin–Schwinger states): positive normalized functionals satisfying the C*-algebraic equilibrium condition at inverse temperature beta. To place this in a motivic framework, Connes, Consani and Marcolli introduced endomotives: an algebraic endomotive over a field K is, roughly, a countable inductive system of finite-dimensional commutative reduced K-algebras — equivalently a projective system of zero-dimensional varieties, i.e. Artin motives — together with a commuting family of endomorphisms sigma_n and partial inverses rho_n; the crossed-product-type algebra built from the limit and the semigroup action encodes both the abelian part and the dynamics. Passing to characters over the algebraic closure and completing yields an analytic endomotive, a C*-algebra carrying a time evolution, and for the Bost–Connes endomotive this recovers the original BC system; the category of endomotives, with (étale) correspondences as morphisms (a correspondence between two varieties is given by an auxiliary variety mapping to both; its noncommutative counterpart is a bimodule), is developed in the Connes–Marcolli book 'Noncommutative Geometry, Quantum Fields, and Motives'. The 'field with one element' F_1 is a hypothetical base over which Spec Z should behave like a curve defined over a finite field; Soulé made this precise through a functor-of-points formalism ('gadgets') in which varieties over F_1 yield ordinary varieties after extension of scalars to Z, with the formal rule F1n⊗F1Z=Z[T]/(Tn−1)\mathbb{F}_{1^n}\otimes_{\mathbb{F}_1}\mathbb{Z}=\mathbb{Z}[T]/(T^n-1) playing the role of the finite extensions of F_1. In 'Fun with F_1', Connes, Consani and Marcolli show that the BC endomotive descends to F_1 — the tower of its abelian part mirrors the extensions F_{1^n}, and the endomorphisms sigma_n realize the analogue of the Frobenius correspondence — and they identify an integral model of the BC-algebra with the integral Hecke algebra. They then study the reduction at a prime p: over a perfect field K of characteristic p ('perfect' means the Frobenius map x maps to x^p is an automorphism), the algebras K[T]/(T^{p^k m}-1) are unreduced Artinian algebras with nilpotents, because Tpk−1=(T−1)pkT^{p^k}-1=(T-1)^{p^k} has multiple roots. The p-primary part of the BC-algebra in characteristic p is nilpotent in nature (it is faithfully represented by infinite lower-triangular matrices); the endomorphism sigma_{p^ell} intertwines with the Frobenius via (σpℓ⊗σFpℓ)(f)=fpℓ(\sigma_{p^\ell}\otimes\sigma_{\mathbb{F}_p}^{\ell})(f)=f^{p^{\ell}}; after functorial reduction the limit algebra becomes the group ring of the prime-to-p part of Q/Z, on which sigma_p becomes an automorphism, and the partial inverses rho_n acquire only denominators prime to p. The authors note that in this positive-characteristic setting the notion of KMS states should be taken in the extended sense of Consani–Marcolli's function-field quantum statistical mechanics (a theory of pointed Drinfeld modules, the function-field analogues of elliptic curves with complex multiplication, whose partition functions are Goss's positive-characteristic L-functions, the function-field analogue of the Riemann zeta function). Crucially, however, the existing definition of endomotives required reduced algebras, in agreement with the classical definition of Artin motives, and the paper also records two subtleties: reducing the abelian part and then forming the crossed product is not the same as modding out the integral crossed-product algebra by its nilpotent radical; and the partial inverses over Z are not ring homomorphisms but only suggest associated correspondences (bimodules). This is what motivates the problem of a general construction of endomotives in arbitrary characteristic.

Problem Statement

Construct a general theory of endomotives valid over (perfect) fields of arbitrary characteristic, extending the characteristic-zero theory of Connes–Consani–Marcolli and reducing to it there. Following the formulation of "Fun with F_1" (the subsection "Endomotives in the unreduced case", which closes its Section 5 on the endomotive in characteristic p), an endomotive over a (perfect) field K should be specified by: (1) an inductive system of augmented commutative K-algebras, finite-dimensional as K-vector spaces — that is, Artinian commutative K-algebras, not assumed reduced, where "augmented" means equipped with a K-algebra map to K (a base point); and (2) a commutative family of correspondences sigma_n; together with the crossed-product-type algebra this data determines. The data must be compatible with the constructions of "Fun with F_1" and of the Connes–Consani–Marcolli theory in the two specific senses stated there: (a) it should determine homomorphisms (correspondences) such as the rho_n whenever denominators (division by n) are allowed — after inverting n the partial inverses must become honest ring homomorphisms, even though over the integers and in characteristic p they exist only as correspondences; and (b) it should be fulfilled by the endomotives associated to self-maps of pointed varieties in the sense of Connes–Consani–Marcolli (their Example 3.4), now without requiring the self-maps to be unramified over the base point. The theory must recover, as canonical instances, the Bost–Connes endomotive in characteristic zero, its integral model, and the analysis of the BC-system over perfect fields of characteristic p. Two companion issues from the same discussion of the paper belong to the same objective and must be settled within the theory: first, a systematic comparison of the two reduction procedures — reducing the abelian part and then forming the crossed product, versus modding out the crossed-product algebra by its nilpotent radical — which the source shows are not equivalent in general, so the theory should determine what each procedure computes and which is the correct reduction for the dynamical and spectral properties; second, the formulation of the thermodynamical notions (time evolution, KMS equilibrium states in the extended sense of the function-field theory of Consani–Marcolli, partition function) for these unreduced positive-characteristic endomotives, and the analysis of whether and how the arithmetic interpretations (relations to zeta and L-functions) persist or must be modified.

The construction is to comprise the definition of the endomotive data, the composition law for the correspondences and the crossed-product-type algebra they determine, and the proofs of the two compatibility requirements (a) and (b); the general framework is to recover the three Bost–Connes instances rather than merely circumscribe them. Of the two companion issues, the comparison of the two reduction procedures is to determine what each procedure computes and which one is the correct reduction for the dynamics and the spectrum; the thermodynamical formulation (time evolution, KMS equilibrium states, partition function) is to be carried out for these unreduced endomotives, together with the analysis of the arithmetic interpretations (the fate of the zeta- and L-function relations).

References

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