Bibliographic record
Abstract
We study the growth of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -primary fine Selmer group, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>′</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , of an elliptic curve over an intermediate sub-extension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic Lie extension, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℒ</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> . We estimate the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -corank of the kernel and cokernel of the restriction map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>ℒ</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>′</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ℒ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">Gal</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ℒ</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>′</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> a finite extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> contained in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℒ</mml:mi> </mml:math> . We show that the growth of the fine Selmer groups in these intermediate sub-extension is related to the structure of the fine Selmer group over the infinite level. On specializing to certain (possibly non-commutative) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic Lie extensions, we prove finiteness of the kernel and cokernel and provide growth estimates on their orders.
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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 | 0.007 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".