STARK POINTS AND -ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Let $E$ be an elliptic curve over $\mathbb{Q}$ , and let ${\it\varrho}_{\flat }$ and ${\it\varrho}_{\sharp }$ be odd two-dimensional Artin representations for which ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$ is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms $f$ , $g$ , and $h$ of respective weights two, one, and one, giving rise to $E$ , ${\it\varrho}_{\flat }$ , and ${\it\varrho}_{\sharp }$ via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain $p$ - adic iterated integrals attached to the triple $(f,g,h)$ , which are $p$ -adic avatars of the leading term of the Hasse–Weil–Artin $L$ -series $L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$ when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on $E$ —referred to as Stark points —which are defined over the number field cut out by ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$ . This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when $g$ and $h$ are binary theta series attached to a common imaginary quadratic field in which $p$ splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing $p$ -adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on ${\mathcal{H}}_{p}\times {\mathcal{H}}$ ), and extensions of $\mathbb{Q}$ with Galois group a central extension of the dihedral group $D_{2n}$ or of one of the exceptional subgroups $A_{4}$ , $S_{4}$ , and $A_{5}$ of $\mathbf{PGL}_{2}(\mathbb{C})$ .
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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 it