Bost–Connes systems, Hecke algebras, and induction
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Bibliographic record
Abstract
We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field K and we show that the C*-algebra of the Bost–Connes system for K can be obtained from our Hecke algebra by induction, from the group of totally positive principal ideals to the whole group of ideals. Our Hecke algebra is therefore a full corner, corresponding to the narrow Hilbert class field, in the Bost–Connes C*-algebra of K ; in particular, the two algebras coincide if and only if K has narrow class number one. Passing the known results for the Bost–Connes system for K to this corner, we obtain a phase transition theorem for our Hecke algebra. In another application of induction we consider an extension L/K of number fields and we show that the Bost–Connes system for L embeds into the system obtained from the Bost–Connes system for K by induction from the group of ideals in K to the group of ideals in L . This gives a C*-algebraic correspondence from the Bost–Connes system for K to that for L . Therefore the construction of Bost–Connes systems can be extended to a functor from number fields to C*-dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMS-states and we show that for \beta>1 certain extremal KMS _\beta -states for L can be obtained, via induction and rescaling, from KMS _{[L: K]\beta} -states for K . On the other hand, for 0<\beta\le1 every KMS _{[L: K]\beta} -state for K induces to an infinite weight.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it