Bibliographic record
Abstract
We consider elliptic curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E slash double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E / \mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the image of the adelic Galois representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript upper E"> <mml:semantics> <mml:msub> <mml:mi> ρ </mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic images, compute all examples of conductor at most 500,000, precisely describe the image of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript upper E"> <mml:semantics> <mml:msub> <mml:mi> ρ </mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.003 | 0.003 |
| Meta-epidemiology (broad) | 0.005 | 0.003 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.020 | 0.056 |
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; both teacher heads agree on what is shown here.
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".