On the failure of pseudo-nullity of Iwasawa modules
Bibliographic record
Abstract
Consider the family of CM-fields which are pro- p p p p -adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z p \mathbf {Z}_p -extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro- p p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p p -adic Lie groups. The proof uses Kida’s formula for the growth of λ \lambda -invariants in cyclotomic Z p {\mathbf Z}_p -extensions of CM-fields. In fact, we give a new proof of Kida’s formula which includes a slight weakening of the usual μ = 0 \mu = 0 assumption. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.007 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".