Energoservis (Czechia)

Sato-Tate families in localized Bost-Connes systems: open phase, no congruence obstruction, and the symmetric power expansion

For a non-CM elliptic curve over Q and a Sato-Tate interval of angles, the associated prime family has positive density, critical exponent one, and an open symmetry-broken phase, with zeta_S(beta) = (delta + o(1)) log(1/(beta-1)) unconditionally, sharpening to O(1) under any effe...

Circle actions on function field Bost-Connes systems: the compact group advantage is illusory, and the Weil zeta function is thermodynamic

The circle action coming from the degree grading on a function field Bost-Connes system is analyzed: the apparent advantage of a compact acting group is shown to be illusory for classification purposes, while the Weil zeta function of the underlying curve is exhibited as a thermo...

Localized GL_2 Bost-Connes systems: the equilibrium range and the automorphic obstruction range are disjoint, by exactly one

For GL_2 the semigroup crossed product is unavailable (the unit group SL_2(Z) is infinite, by the unit-group exclusion of Paper X) and one must pass to the Connes-Marcolli Hecke quotient, whose local factor is the Solomon zeta of M_2(Z_p), computed by the sublattice count sigma(p...

The Galois-twisted arithmetic groupoid: its KMS classification, and why the Tannakian obstruction does not compute it

The Galois-twisted groupoid model, in which the Frobenius elements act through a homomorphism of the abelian ideal group (hence with commuting lifts), is classified: its KMS_beta simplex is Prob(E_beta \ G) for the abelian essential-value group E_beta, which agrees with the Tanna...

Isogeny, zeta-equivalence, and the Picard torsor in function field Bost-Connes systems

For function field Bost-Connes systems the deep stratum of the boundary is a torsor under the Picard group, and under the full gauge action of Paper XIV the closure of the translation monoid in its homeomorphism group is a compact group acting simply transitively, recovering the...

The statistical mechanics of sieve-theoretic primes: the critical spectrum, the log-power dichotomy, and two families that are never unique

For six sieve-theoretic prime families the critical exponents equal the counting exponents, and the open/closed character of the symmetry-broken phase is the power of the logarithm, hence the number of simultaneous prime conditions. Confinement to a single residue class obstructs...

Localized Bost-Connes systems in arithmetic topology: local class field theory, linking Kakutani dichotomies, and topological phase transitions

The arithmetic-topology dictionary (primes as knots, fields as three-manifolds) is transported to localized Bost-Connes systems. The Gauss linking matrix formula (Theorem 3.2) expresses the obstruction data through linking numbers via local class field theory, and the meridian mo...

Leopoldt and the p-local Bost-Connes groupoid: the rank of the isotropy, not its triviality, and why log_p(Np) vanishes

The isotropy of the p-local Bost-Connes groupoid at an intermediate stratum Y_T is identified exactly: it is the kernel of the map from the lattice of T-supported principal divisors to the quotient of the complementary unit factors by the projected closure of the global units. Co...

Localized Bost-Connes systems IV: noncommutative metric geometry

The localized Bost-Connes algebra is made a compact quantum metric space. Lipschitz seminorms adapted to the prime coordinates are constructed, the tensorization of spectral gaps across primes is controlled (Lemma 6.1), the exclusivity theorem shows the tail observables lie outsi...

Localized Bost-Connes systems III: invariants, entropy, and the measure class

This paper isolates which invariants of a localized Bost-Connes system depend on the inverse temperature and which do not. The measurable cohomology of the tail relation with circle coefficients is computed through its kernel, the beta-dependence is located exactly (Corollary 2.4...

Circle actions on function field Bost-Connes systems: the compact group advantage is illusory, and the Weil zeta function is thermodynamic

The circle action coming from the degree grading on a function field Bost-Connes system is analyzed: the apparent advantage of a compact acting group is shown to be illusory for classification purposes, while the Weil zeta function of the underlying curve is exhibited as a thermo...

A non-commutative Ore LCM semigroup with arithmetic thermodynamics: the Hurwitz quaternions

The Hurwitz quaternion order provides a genuinely non-commutative right Ore, right LCM monoid whose Nica-Toeplitz system has arithmetic thermodynamics, with partition function zeta(beta) zeta(beta-1) and critical exponent 2. Conditions (C1)-(C4) for a non-injective arithmetic Bos...

Leopoldt and the p-local Bost-Connes groupoid: the rank of the isotropy, not its triviality, and why log_p(Np) vanishes

The isotropy of the p-local Bost-Connes groupoid at an intermediate stratum Y_T is identified exactly: it is the kernel of the map from the lattice of T-supported principal divisors to the quotient of the complementary unit factors by the projected closure of the global units. Co...

Localized GL_2 Bost-Connes systems: the equilibrium range and the automorphic obstruction range are disjoint, by exactly one

For GL_2 the semigroup crossed product is unavailable (the unit group SL_2(Z) is infinite, by the unit-group exclusion of Paper X) and one must pass to the Connes-Marcolli Hecke quotient, whose local factor is the Solomon zeta of M_2(Z_p), computed by the sublattice count sigma(p...

Towards non-abelian localized Bost-Connes systems: the matrix Kakutani dichotomy and a Tannakian obstruction

For a compact group G with Frobenius classes indexed by a set S of primes, the obstruction series Xi_beta(rho,S) = sum over p of N(p)^{-beta} ||I - rho(Frob_p)||_HS^2 defines a Tannakian category T_beta = Rep(G/N_beta) (the matrix Kakutani dichotomy, for commuting lifts), with th...

The Galois-twisted arithmetic groupoid: its KMS classification, and why the Tannakian obstruction does not compute it

The Galois-twisted groupoid model, in which the Frobenius elements act through a homomorphism of the abelian ideal group (hence with commuting lifts), is classified: its KMS_beta simplex is Prob(E_beta \ G) for the abelian essential-value group E_beta, which agrees with the Tanna...

Localized Bost-Connes systems V: the free variant and the free critical exponent

The free-probabilistic variant of the localized Bost-Connes construction is built, replacing the tensor independence of the prime coordinates by free independence. The free critical exponent is identified and compared with the classical one, and the extent to which the phase-tran...

The statistical mechanics of sieve-theoretic primes: the critical spectrum, the log-power dichotomy, and two families that are never unique

For six sieve-theoretic prime families the critical exponents equal the counting exponents, and the open/closed character of the symmetry-broken phase is the power of the logarithm, hence the number of simultaneous prime conditions. Confinement to a single residue class obstructs...

Localized Bost-Connes systems V: the free variant and the free critical exponent

The free-probabilistic variant of the localized Bost-Connes construction is built, replacing the tensor independence of the prime coordinates by free independence. The free critical exponent is identified and compared with the classical one, and the extent to which the phase-tran...

Isogeny, zeta-equivalence, and the Picard torsor in function field Bost-Connes systems

For function field Bost-Connes systems the deep stratum of the boundary is a torsor under the Picard group, and under the full gauge action of Paper XIV the closure of the translation monoid in its homeomorphism group is a compact group acting simply transitively, recovering the...