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Bibliographic record
Abstract
Given a number field extension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K slash k"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K/k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with an intermediate field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript plus"> <mml:semantics> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">K^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fixed by a central element of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G a l left-parenthesis upper K slash k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Gal</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Gal}(K/k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of prime order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there exists an algebraic torus over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose rational points are elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript times"> <mml:semantics> <mml:msup> <mml:mi>K</mml:mi> <mml:mo> × </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">K^\times</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Superscript times"> <mml:semantics> <mml:msup> <mml:mi>k</mml:mi> <mml:mo> × </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">k^\times</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the norm map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript upper K slash upper K Sub Superscript plus"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">N_{K/K^+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K slash k"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K/k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Galois. Partial results including the numerator of Ono’s formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper K colon upper K Superscript plus Baseline right-bracket equals 2"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>K</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">[K:K^+]=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a CM-field. This case corresponds to maximal tori in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper S normal p Subscript 2 n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</m
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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 | 0.001 |
| 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 | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 it