The Modularity of Special Cycles on Orthogonal Shimura Varieties over Totally Real Fields under the Beilinson–Bloch Conjecture
Bibliographic record
Abstract
Abstract We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree d associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at e real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e . Recently, these cycles were constructed by Kudla and Rosu–Yott, and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson–Bloch conjecture on the injectivity of the higher Abel–Jacobi map, the generating series of special cycles of codimension er in the Chow group is a Hilbert–Siegel modular form of genus r and weight $1+n/2$ . Our result is a generalization of Kudla’s modularity conjecture, solved by Yuan–Zhang–Zhang unconditionally when $e=1$ .
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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".