Modular algorithms for Gross–Stark units and Stark–Heegner points
Bibliographic record
Abstract
In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic logarithms of Gross–Stark units and Stark–Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross–Stark units and Stark–Heegner points from them. Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Gross–Stark units in narrow Hilbert class fields of real quadratic fields with discriminants up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10000"> <mml:semantics> <mml:mn>10000</mml:mn> <mml:annotation encoding="application/x-tex">10000</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for primes less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="20"> <mml:semantics> <mml:mn>20</mml:mn> <mml:annotation encoding="application/x-tex">20</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , as well as Stark–Heegner points on elliptic curves.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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; both teacher heads agree on what is shown here.
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".