On the properties of Northcott and Narkiewicz for elliptic curves
Bibliographic record
Abstract
In this paper, as an analog of the number field case, for an elliptic curve [Formula: see text] defined over the algebraic numbers and for any subfield [Formula: see text] of algebraic numbers, we say that [Formula: see text] has the Northcott property over [Formula: see text] if there are at most finitely many [Formula: see text]-rational points on [Formula: see text] of uniformly bounded height, and we say that [Formula: see text] has the property (P) over [Formula: see text] if for any infinite subset [Formula: see text] of [Formula: see text]-rational points on [Formula: see text], [Formula: see text] for an [Formula: see text]-endomorphism [Formula: see text] of [Formula: see text] implies that [Formula: see text] is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P), but the converse is not true.
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How this classification was reachedexpand
Full frame distilled prediction
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".