Bibliographic record
Abstract
Let G be a discrete subgroup of PSL(2, R), and consider closed geodesics on the quotient of the upper half plane by G. In this thesis, we study how these geodesics intersect when G is the group of units of norm 1 in an Eichler order of an indefinite quaternion algebra over Q, i.e. over a Shimura curve. We start by developing some theory for a general G, providing a pathway to interpret the geometry of the situation (intersection point, angle, etc.) in terms of the algebra of G. The next focus is on G=PSL(2, Z), where we relate the work to Conway's topograph, as well as work by Duke, Imamoglu, and Toth on the linking number of modular knots. We then detail some background on quaternion algebras, before diving in to the Shimura curve case. We re-interpret intersection numbers in terms of x-linking of optimal embeddings of real quadratic orders, and produce formulas which count the total amount of x-linking in a given Eichler order. These formulas are essentially a real quadratic analogue to classical results of Gross and Zagier. We define Hecke operators acting on optimal embeddings, and using the signed intersection pairing, produce formal q-series, which are proven to be weight two modular forms on Gamma_0(N). The thesis is concluded with a conjectural relation to work of Darmon and Vonk on a real quadratic analogue to the difference of j-values, a brief description of the practical algorithms used to compute with intersection numbers, and a short survey of future projects
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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; 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".