Group Structures of Elliptic Curves Over Finite Fields
Bibliographic record
Abstract
It is well known that if E is an elliptic curve over the finite field , then for some positive integers m,k. Let S(M,K) denote the set of pairs (m,k) with m≤M and k≤K for which there exists an elliptic curve over some prime finite field whose group of points is isomorphic to . Banks, Pappalardi, and Shparlinski recently conjectured that if , then a density zero proportion of the groups in question actually arises as the group of points on some elliptic curve over some prime finite field. On the other hand, if , they conjectured that a density 1 proportion of the groups in question arises as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range , and we prove that the second part of their conjecture holds in the limited range K≥M4+ϵ. In the wider range K≥M2, we show that at least a positive density of the groups in question actually occurs.
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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 | 0.007 |
| 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 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.011 | 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".