A random matrix model for elliptic curve<i>L</i>-functions of finite conductor
Bibliographic record
Abstract
We propose a random-matrix model for families of elliptic curve L -functions of finite conductor. A repulsion of the critical zeros of these L -functions away from the centre of the critical strip was observed numerically by Miller (2006 Exp. Math. 15 257–79); such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random-matrix model for Miller’s surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this sub-ensemble of SO (2 N ) the excised orthogonal ensemble. The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of L -functions implied by the formulae of Waldspurger and Kohnen–Zagier. The cut-off scale appropriate to modelling elliptic curve L -functions is exponentially small relative to the matrix size N . The one-level density of the excised ensemble can be expressed in terms of that of the well-known Jacobi ensemble, enabling the former to be explicitly calculated. It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of SO (2 N ). When N → ∞ we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller’s discrepancy. Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".