Normal functions, Picard--Fuchs equations, and elliptic fibrations on<i>K</i>3 surfaces
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Bibliographic record
Abstract
Abstract Using Gauss–Manin derivatives of generalized normal functions, we arrive at results on the non-triviality of the transcendental regulator for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> $K_{m}$ of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> $K_{1}$ of a very general <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo></m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> $K3$ surface and its self-product. We also construct an explicit family of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> $K_{1}$ cycles on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>⊕</m:mo> <m:msub> <m:mi>E</m:mi> <m:mn>8</m:mn> </m:msub> <m:mo>⊕</m:mo> <m:msub> <m:mi>E</m:mi> <m:mn>8</m:mn> </m:msub> </m:mrow> </m:math> $H\oplus E_{8}\oplus E_{8}$ -polarized <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo></m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> $K3$ surfaces, and show they are indecomposable by a direct evaluation of the real regulator. Critical use is made of natural elliptic fibrations, hypersurface normal forms, and an explicit parametrization by modular functions.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it