Bibliographic record
Abstract
Abstract Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb {Q}$ has open image, and in particular, there is a minimal natural number $C_E$ such that the mod $\ell $ representation ${\bar {\rho }}_{E,\ell }$ is surjective for any prime $\ell> C_E$ . Assuming the Generalized Riemann Hypothesis, Mayle–Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of E and have explicit constants. The method is based on using effective forms of the Chebotarev Density Theorem together with the Faltings–Serre method, in particular, using the “deviation group” of the $2$ -adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the $2$ -adic representation $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article how to further reduce the constants in Mayle–Wang’s results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.
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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 | 0.004 | 0.015 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.005 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.012 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
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".