Hecke Operators on Vector-Valued Modular Forms
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Bibliographic record
Abstract
We study Hecke operators on vector-valued modular forms for the Weil representation L of a lattice L. We first construct Hecke operators T r that map vector-valued modular forms of type L into vector-valued modular forms of type L(r) , where L(r) is the lattice L with rescaled bilinear form (, ) r = r(, ), by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vectorvalued Hecke operators T r have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators T r . In the particular case when r = n 2 for some positive integer n, we compose T n 2 with a projection operator to construct new Hecke operators H n 2 that map vector-valued modular forms of type L into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators H n 2 , and compare our operators with the alternative construction of Bruinier-Stein [Math.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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