Computing elliptic curves over ℚ via Thue-Mahler equations and related problems
Bibliographic record
Abstract
We present a practical and efficient algorithm for solving an arbitrary Thue-Mahler equation. This algorithm uses explicit height bounds with refined sieves, combining Diophantine approximation techniques of Tzanakis-de Weger with new geometric ideas. We begin by using methods of algebraic number theory to reduce the problem of solving the Thue-Mahler equation to the problem of solving a finite collection of related Diophantine equations. In the first part of this thesis, we establish the key results which allow us to drastically reduce the number of such Diophantine equations and subsequently reduce the running time. In the second part of this thesis, we show that, by fixing one exponent, there exists an effectively computable constant bounding the solutions of a Goormaghtigh equation under certain conditions. For small values of this fixed exponent, we solve the equation completely. For one such small exponent, we modify and specialize our Thue-Mahler algorithm to the resulting equation in order to fully resolve this case. In the third part, we discuss an algorithm for finding all elliptic curves over ℚ with a given conductor. Though based on classical ideas derived from reducing the problem to one of solving associated Thue-Mahler equations, our approach, in many cases at least, appears to be reasonably efficient computationally. We provide details of the output derived from running the algorithm, concentrating on the cases of conductor p or p², for p prime, with comparisons to existing data. Finally, we specialize the Thue-Mahler algorithm to degree 3, applying an analogue of Matshke-von Kanel’s elliptic logarithm sieve to construct a global sieve, leading to reduced search spaces. The algorithm is implemented in the Magma computer algebra system, and is part of an ongoing collaborative project.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".