Numerical Verification of a “Birch and Swinnerton-Dyer Type ” Conjecture
Bibliographic record
Abstract
In [Dar92], Darmon gave a description of a “Birch and Swinnerton-Dyer ” type conjecture attached to a modular elliptic curve E defined over the rational numbers and to a quadratic field. A theta object is constructed using Heegner points and cycles for those curves, and can be shown to interpolate special values of L-functions through formulas of Gross-Zagier and Waldspurger. The conjecture relates the “leading coefficient ” of this theta object to the arithmetic data of the curve, in particular through a regulator given by a height pairing described by Mazur and Tate in [MT87]. In [Dar92], the lead-ing coefficient for this theta element was calculated numerically for several cases, in particular for the modular curve of conductor 37 and many real quadratic number fields. Our goal is to compute the regulator in those cases in order to verify the conjecture. Along the way, we outline a procedure to calculate the Mazur-Tate height pairing in practice. I
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".