Quadratic forms of holomorphic cusp forms and the decay of their $\ell^p$-norms for $0 < p < 2$
Bibliographic record
Abstract
In this paper, we demonstrate that, given an orthonormal basis of holomorphic Hecke cusp forms, conditionally, quadratic forms composed of cusp forms -- each expressed as a bounded linear combination of holomorphic Hecke cusp forms -- are generally not themselves expressible as bounded linear combinations of holomorphic Hecke cusp forms when the sum of the weights exceeds some absolute constant, provided that the coefficients of the quadratic form satisfy appropriate nonvanishing and boundedness conditions. This illustrates the finiteness of the number of solutions to the linear equation of modular forms equated to a quadratic form of large weight. We also show that, conditionally, for $0 < p < 2$, the $\ell^p$-norm of such quadratic forms in holomorphic Hecke cusp forms tends to zero asymptotically with respect to expansion in this orthonormal basis of Hecke eigenforms.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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".