The arithmetic of vector-valued modular forms on Γ0(2)
Bibliographic record
Abstract
Let [Formula: see text] denote an irreducible two-dimensional representation of [Formula: see text] The collection of vector-valued modular forms for [Formula: see text], which we denote by [Formula: see text], form a graded and free module of rank two over the ring of modular forms on [Formula: see text], which we denote by [Formula: see text] For a certain class of [Formula: see text], we prove that if [Formula: see text] is any vector-valued modular form for [Formula: see text] whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of [Formula: see text] is unbounded. Our methods involve computing an explicit basis for [Formula: see text] as a [Formula: see text]-module. We give formulas for the component functions of a minimal weight vector-valued form for [Formula: see text] in terms of the Gaussian hypergeometric series [Formula: see text], a Hauptmodul of [Formula: see text], and the Dedekind [Formula: see text]-function.
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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 | 0.001 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".