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 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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 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 teacher head, 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".