Bibliographic record
Abstract
Abstract. This note presents a connection between Ulmer’s construction [Ulm02] of non-isotrivial elliptic curves over Fp(t) with arbitrarily large rank, and the theory of Heegner points (attached to parametrisations by Drinfeld modular curves, as sketched in Section 3 of Ulmer’s article (see page??). This ties in the topics in Section 4 of that article more closely to the main theme of this volume. A review of the number field setting. Let K be a quadratic imaginary extension of F = Q, and let E /Q be an elliptic curve of conductor N. When all the prime divisors of N are split in K/F, the Heegner point construction (in the most classical form that is considered in [GZ], relying on the modular parametrisation X0(N) − → E) produces not only a canonical point on E(K), but also a norm-coherent system of such points over all abelian extensions of K which are of “dihedral type”. (An abelian extension H of K is said to be of dihedral type if it is Galois over Q and the generator of Gal(K/Q) acts by −1 on the abelian normal subgroup Gal(H/K).) The existence of this construction is consistent with the Birch and Swinnerton-Dyer conjecture, in the following sense: an analysis of the sign in the functional equation for L(E/K, χ, s) = L(E/K, ¯χ, s) shows that this sign is always equal to −1, for all complex characters χ of G:= Gal(H/K). Hence The product factorisation implies that L(E/K, χ, 1) = 0 for all χ: G − → C ×. L(E/H, s) = ∏ L(E/K, χ, s) χ
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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 | 0.000 | 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.020 | 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".