Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences
Bibliographic record
Abstract
Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".