Functoriality for the exterior square of GL₄ and the symmetric fourth of GL₂
Bibliographic record
Abstract
In this paper we prove the functoriality of the exterior square of cusp forms on G L 4 GL_{4} as automorphic forms on G L 6 GL_{6} and the symmetric fourth of cusp forms on G L 2 GL_{2} as automorphic forms on G L 5 GL_{5} . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain L L -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of G L 4 GL_{4} and the absolute convergence of the exterior square L L -functions of G L 4 GL_{4} . Second, we prove that the fourth symmetric power L L -functions of cuspidal representations of G L 2 GL_{2} are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound 3 26 \frac {3}{26} for Hecke eigenvalues of Maass forms over any number field.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.014 | 0.003 |
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".