On the integrals of the Kudla-Millson theta series
Bibliographic record
Abstract
The Kudla-Millson theta series &thetas;km of a pseudoeuclidean space V of signature (p, q) and lattice L is a differential form on the symmetric space D attached to the pseudoorthogonal group O(p, q) that transforms like a genus n Siegel modular form of weight (p + q)/2. Any integral of &thetas;km inherits the modular transformation law and becomes a nonholomorphic Siegel modular form. A special case of such integral is the well-known Zagier Eisenstein series F(τ) of weight 3/2 as showed by Funke. We show that for n = 1 and p = 1 the integral of &thetas;km along a geodesic path coincides with the Zwegers theta function Θa,b. We construct a higher-dimensional generalization of Zwegers theta functions as integrals of &thetas;km over geodesic simplices for n ≥ 2. If Γ is a discrete group of isometries of V that preserve the lattice L and act trivially on the cosets L*/L, then the fundamental region Γ\D is an arithmetic locally symmetric space. We prove that the integral of &thetas; km over Γ\D converges and compute it in some cases. In particular, we extend the results of Kudla to the cases p = 1, and q odd.
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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 | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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 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".