Towards the Sato–Tate groups of trinomial hyperelliptic curves
Bibliographic record
Abstract
We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form [Formula: see text] where [Formula: see text] is the genus of the curve and [Formula: see text] is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for [Formula: see text] curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus [Formula: see text] curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components [Formula: see text] for these families of curves.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.007 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".