Stark–Heegner points and the Shimura correspondence
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Bibliographic record
Abstract
Abstract Let $g = \sum c(D)q^D$ and $f=\sum a_n q^n$ be modular forms of half-integral weight k +1/2 and integral weight 2 k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D , a theorem of Waldspurger relates the coefficient c ( D ) to the central critical value L ( f , D , k ) of the Hecke L -series of f twisted by the quadratic Dirichlet character of conductor D . This paper establishes a similar kind of relationship for central critical derivatives in the special case k =1, where f is of weight 2. The role of c ( D ) in our main theorem is played by the first derivative in the weight direction of the D th Fourier coefficient of a p -adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p -adic deformations of modular forms.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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