Factorization Tests and Algorithms Arising from Counting Modular Forms and Automorphic Representations
Bibliographic record
Abstract
Abstract A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ to a simpler function of $k$ and $N$ , showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ . It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found not to be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$ , then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.
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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 | 0.003 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 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".