The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems
Bibliographic record
Abstract
The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> - conjecture for Galois extensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> of number fields. It is certainly more difficult than the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ℚ</mml:mi> </mml:mrow> </mml:math> and the degree of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important point is the following: While dealing with our Euler systems, we have to keep track of the action of the Galois group, whose order is not invertible in the coefficient ring <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> . At the end we prove a generalization of the well-known Rédei-Reichardt theorem and explain the close link with our theory.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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".