REDUCTION modp OF SUBGROUPS OF THE MORDELL–WEIL GROUP OF AN ELLIPTIC CURVE
Bibliographic record
Abstract
Let E be an elliptic curve defined over ℚ. Let Γ be a free subgroup of rank r of E(ℚ). For any prime p of good reduction, let Γ p be the reduction of Γ modulo p and E p be the reduction of E modulo p. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, [Formula: see text] where ∊(p) is any function of p such that ∊(p) → 0 as p → ∞. This is a consequence of two other results. Denote by N p the cardinality of E p (𝔽 p ), where 𝔽 p is a finite field of p elements. Then for any δ > 0, the set of primes p for which N p has a divisor in the range (p δ - ∊(p) , p δ + ∊(p) ) has density zero. Moreover, the set of primes p for which [Formula: see text] has density zero.
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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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".