Juwen Lie (engineer, not a professional mathematician), working with an AI assistant. Last updated 2026-11-07.
For odd squarefree conductor N and primes P with 0 < P/N ≤ 0.25 we compared the average of ε·a2·aP over elliptic curves (ε = root number) with the same average over all weight-2 newforms of level N. The newform side was computed exactly with the Eichler–Selberg trace formula for tr(T2P WN) (a classical theorem; our implementation matched the elliptic-curve sums at all 180/180 small-level checks). The table gives the elliptic-curve average divided by the newform average (1 = same scale).
| N range | curves | ε·a2·aP ratio | plain ε·aP ratio | difference |
|---|---|---|---|---|
| 2000–3999 (discovery) | 1005 | 0.96 [0.84, 1.08] | 0.67 [0.55, 0.79] | [0.13, 0.46] |
| 4000–7999 (pre-registered replication) | 1973 | 1.007 [0.91, 1.10] | 0.495 [0.40, 0.59] | [0.37, 0.64] |
Observation: weighting by a2 brings the elliptic-curve murmuration to the newform scale (ratio ≈ 1), while the plain murmuration stays well below it. All three pre-registered criteria passed in the replication range. Novelty: this looks like a 2-adic local effect and is plausibly explained by the local-factor heuristics of Sawin and Sutherland (arXiv:2504.12295, §1.5). We do not claim it as new. A direct test of their prime-conductor prediction with 419 curves was underpowered (ratio 0.62 [0.42, 0.84], shape correlation 0.84): not established either way.
We split curves by how they behave at the prime 2 (supersingular with even or odd N/2-part, additive at 2, etc.) and compared their murmuration scales. The hypotheses and pass/fail thresholds were registered in the research log before the 20,000–29,999 data were downloaded (44,141 isogeny classes; 957,560 ap values checked against Cremona's aplist files with 0 mismatches). One metric was found to be ill-defined before the run and was replaced by a documented amendment.
| pre-registered test | ratio [95% CI] | rule | result |
|---|---|---|---|
| H1: additive-odd vs supersingular-even | 0.956 [0.904, 1.003] | CI contains 1, shape corr ≥ 0.9 (0.983) | pass |
| H2: supersingular-odd vs supersingular-even | 0.408 [0.338, 0.474] | CI below 1 | pass |
| H3′: supersingular-odd vs additive-odd | 0.605 [0.534, 0.674] | point ≤ 0.60 | fail |
Verdict: not replicated under the registered rule (H3′ missed its threshold by 0.005). We report it as a failure rather than adjusting the threshold. A secondary, also pre-registered prediction — a square-root “kink” at P/N = 1/16 present only for 4 | N — was confirmed on the unseen data: kink coefficient −4.49 [−4.78, −4.21] for the 4 | N class and 0.10 [−0.20, 0.38] for the control class.
Interpretation (exploratory): an exact all-newform average computed from the general Eichler–Selberg / Fricke trace formula (following arXiv:1706.02691, checked against elliptic-curve data at 124 levels, 91 of them non-squarefree) reproduces the same ordering of the classes, and its closed form a√y − b·y (+ c√(16y−1) when 4 | N) explains the kink. So most of the effect is already in the trace formula; the remaining elliptic-curve-only dip for the supersingular-odd class is unexplained.
Using ~300,000 consecutive zeros per height from the LMFDB (Platt's database), we measured how far the nearest-neighbour spacing distribution is from the GUE limit and compared it with the first-order correction of Bogomolny, Bohigas, Leboeuf and Monastra (BBLM), which predicts an amplitude A = 1 in units of 1/Neff2.
| height t | Neff | A (BBLM = 1) | 95% CI |
|---|---|---|---|
| 1e8 | 3.82 | 0.65 | [0.60, 0.71] |
| 2e8 | 3.98 | 0.67 | [0.61, 0.74] |
| 5e8 | 4.19 | 0.69 | [0.62, 0.76] |
| 1e9 | 4.35 | 0.70 | [0.62, 0.78] |
| 2e9 | 4.51 | 0.72 | [0.64, 0.81] |
| 5e9 | 4.72 | 0.73 | [0.63, 0.83] |
| 1e10 | 4.88 | 0.71 | [0.60, 0.82] |
| 2e10 | 5.04 | 0.77 | [0.64, 0.88] |
| 3e10 | 5.13 | 0.76 | [0.64, 0.88] |
Observation (post-hoc, exploratory): A rises slowly from about 0.65 at t = 108 to about 0.77 at t = 3·1010, still below 1. This is consistent with the BBLM first-order term being asymptotically correct with a sizeable higher-order correction. Two simple scaling laws fitted at lower heights were rejected at all four of the original LMFDB heights. The LMFDB holds no zeros above about 3·1010, so the trend cannot be followed further with this data. Higher-order corrections are discussed by Bogomolny et al. and by Forrester and Mays; we have not checked whether these numbers are new.
Elliptic-curve data: Cremona's elliptic-curve tables (allcurves files; conductors below 10,000 for section 1, 20,000–29,999 for section 1b). Zeros: LMFDB. Bootstrap: resampling curves (BSD) and blocks of 500 spacings (RH), 2,000 draws. The code is available on request.
Contact: via the GitHub profile juw12.