ON MODULAR GALOIS REPRESENTATIONS MODULO PRIME POWERS
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Bibliographic record
Abstract
We study modular Galois representations mod p m . We show that there are three progressively weaker notions of modularity for a Galois representation mod p m : We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a level-lowering result ("stripping-of-powers of p away from the level"): A mod p m strongly modular representation of some level N p r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p m to any "dc-weak" eigenform, and hence to any eigenform mod p m in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general.
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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 | 0.002 |
| 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 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.010 | 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 it