A theta operator on Picard modular forms modulo an inert prime
Bibliographic record
Abstract
Serre [34] introduced a certain differential operator θ on (elliptic) modular forms over Fp .In terms of the q-expansion f = ∞ n=0 a n q n (0.1) (a n ∈ Fp ) of such a form, θ is given by qd/dq.It lifts, by the same formula, to the space of p-adic modular forms.This suggests a relation with the Tate twist of the mod p Galois representation attached to f , if the latter is a Hecke eigenform.Over C, this operator has been considered already by Ramanujan, where it fails to preserve modularity "by a multiple of E 2 ".Maass modified it so that modularity is preserved, sacrificing holomorphicity.Shimura studied Maass' differential operators on more general symmetric domains, as well as their iterations.They have become known as Maass-Shimura operators and play an important role in the theory of automorphic forms [37, chapter III].At the same time, Serre's p-adic operator has been studied in relation to mod p Galois representations, congruences between modular forms, p-adic families of modular forms and p-adic L-functions.As an example, we cite Coleman's celebrated classicality theorem, asserting that "overconvergent modular forms of small slope are classical" [6].A key step in Coleman's original proof of that theorem was the observation that, although the padic theta operator did not preserve the space of overconvergent modular forms, for any k ≥ 0, θ k+1 mapped overconvergent forms of weight -k to overconvergent forms of weight k + 2.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.007 | 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".