Nonvanishing of Central Values of <i>L</i>-functions of Newforms in S<sub>2</sub>(Γ<sub>0</sub>(dp<sup>2</sup>)) Twisted by Quadratic Characters
Bibliographic record
Abstract
Abstract We prove that for d ∊ {2, 3, 5, 7, 13} and K a quadratic (or rational) field of discriminant D and Dirichlet character 𝜒 , if a prime p is large enough compared to D , there is a new form f ∊ S 2 Γ 0 ( dp 2 )) with sign (+1) with respect to the Atkin–Lehner involution wp 2 such that L ( f ⊗ 𝜒, •)≠ 0. This result is obtained through an estimate of a weighted sum of twists of L -functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the L -functions L ( f ⊗ 𝜒, · ) and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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; both teacher heads agree on what is shown here.
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".