Bibliographic record
Abstract
For a quartic non-biquadratic CM field K, we say that a rational prime p is evil forK if at least one of the principally polarized abelian varieties with CM by K reduces modulo a prime ideal p|p to a product of supersingular elliptic curves with the product polarization. We showed that for fixed K, such primes are bounded by a quantity related to the discriminant of K. We show that evil primes are ubiquitous in the sense that for any rational prime p, there are an infinite number of such CM fields K for which p is evil. (Assuming a standard conjecture, the result holds for a finite set of primes simultaneously.) The proof consists of two parts: (1) showing the surjectivity of the principally polarized abelian varieties with CM by K, for K satisfying some conditions, onto the superspecial points of the reduction modulo p of the Hilbert modular variety associated to the intermediate real quadratic field of K, and (2) showing the surjectivity of the superspecial points of the reduction modulo p of the Hilbert modular variety associated to a real quadratic field with large enough discriminant onto the superspecial points on the reduction modulo p of the Siegel moduli space parameterizing abelian surfaces with principal polarization.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".