Bibliographic record
Abstract
Consider A an abelian variety of dimension r defined over Q. Assume that rankQA≥g, where g≥0 is an integer, and let a1,…,ag∈A(Q) be linearly independent points. (So, in particular, a1,…,ag have infinite order, and if g=0, then the set {a1,…,ag} is empty.) For p a rational prime of good reduction for A, let A¯ be the reduction of A at p, let a¯i for i=1,…,g be the reduction of ai (modulo p), and let 〈a¯1,…,a¯g〉 be the subgroup of A¯(Fp) generated by a¯1,…,a¯g. Assume that Q(A[2])=Q and Q(A[2],2−1a1,…,2−1ag)≠Q. (Note that this particular assumption Q(A[2])=Q forces the inequality g≥1, but we can take care of the case g=0, under the right assumptions, also.) Then in this article, in particular, we show that the number of primes p for which A¯(Fp)〈a¯1,…,a¯g〉 has at most (2r−1) cyclic components is infinite. This result is the right generalization of the classical Artin’s primitive root conjecture in the context of general abelian varieties; that is, this result is an unconditional proof of Artin’s conjecture for abelian varieties. Artin’s primitive root conjecture (1927) states that, for any integer a≠±1 or a perfect square, there are infinitely many primes p for which a is a primitive root (modp). (This conjecture is not known for any specific a.)
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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".