Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two
Bibliographic record
Abstract
Abstract Let V be an algebraic K3 surface defined over a number field K. Suppose V has Picard number two and an infinite group of automorphisms A = Aut(V/K). In this paper, we introduce the notion of a vector height h: V → Pic(V) ⊗ and show the existence of a canonical vector height with the following properties: where σ ∈ A, σ* is the pushforward of σ (the pullback of σ−1), and hD is a Weil height associated to the divisor D. The bounded function implied by the O(1) does not depend on P. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an A-orbit satisfies Here, μ(P) is a nonnegative integer, s is a positive integer, and ω is a real quadratic fundamental unit.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), 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".