Abelian surfaces over finite fields with prescribed groups
Bibliographic record
Abstract
Let A be an abelian surface over F q , the field of q elements. The rational points on A / F q form an abelian group A ( F q ) ≃ Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z . We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z do not occur if n 1 is very large with respect to n 2 , n 3 , n 4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| 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 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".