On the distribution of the number of points on algebraic curves in extensions of finite fields
Bibliographic record
Abstract
Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over Fq, the finite field of q elements.Let #C(F q n ) be the number of F q n -rational points on C.Under a certain multiplicative independence condition on the roots of the zetafunction of C, we derive an asymptotic formula for the number of n = 1, . . ., N such that (#C(F q n ) -q n -1)/2gq n/2 belongs to a given interval I ⊆ [-1, 1].This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve E is defined over Q and considered modulo consecutive primes p, although in our scenario the distribution function is different.The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings.It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus g = 2.
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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 | 0.007 | 0.051 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.005 | 0.002 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".