On endomorphism algebras of $\textup{GL}_{2}$-type abelian varieties and Diophantine applications
Bibliographic record
Abstract
Let f and g be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that an isomorphism between the residual Galois representations attached to f and to g for a large prime p (depending only on g ) implies that the endomorphism algebra of the abelian variety A_{f} , attached to f by the Eichler–Shimura construction (after tensoring with \mathbb{Q} ), is a subalgebra of the endomorphism algebra of the abelian variety A_{g} attached to g . This implies important relations between their building blocks. A non-trivial application of our result is that for all prime numbers d congruent to 3 modulo 8 satisfying that the class number of \mathbb{Q}(\sqrt{-d}) is prime to 3 , the equation x^{4}+dy^{2} =z^{p} has no non-trivial primitive solutions when p is large enough. We prove a similar result for the equation x^{2}+dy^{6}=z^{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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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)
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".