Bibliographic record
Abstract
Let E be an elliptic curve over ℚ. We know that the ring of endomorphisms of its reduction modulo an ordinary prime p is an order of the quadratic imaginary field generated by the Frobenius element π p . However, except in the trivial case of complex multiplication, very little is known about the fields that appear as algebras of endomorphisms when p varies. In this paper, we study the endomorphism ring by looking at the arithmetic of [Formula: see text], the discriminant of the characteristic polynomial of π p . In particular, we give a precise asymptotic for the function counting the number of primes p up to x such that [Formula: see text] is square-free and in certain congruence class fixed a priori, when averaging over elliptic curves defined over the rationals. We discuss the relation of this result with the Lang–Trotter conjecture, and some other questions on the curve modulo p.
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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 | 0.003 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".