Hecke Operators on Vector-Valued Modular Forms
Bibliographic record
Abstract
We study Hecke operators on vector-valued modular forms for the Weil representation ρ L of a lattice L. We first construct Hecke operators T r that map vector-valued modular forms of type ρ L into vector-valued modular forms of type ρ 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 ρ 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.Z. 264 (2010), 249-270] and Stein [Funct.Approx.Comment.Math.52 (2015), 229-252].
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".