The covariance between Frobenius traces and L-function values changes sign.
Wachs (arXiv:2603.22807) investigates murmurations — the recently discovered oscillatory patterns in families of elliptic curves — by decomposing them into contributions from local factors. Over function fields, murmuration densities reduce to type-weighted densities with no within-type zero displacement. The oscillations come entirely from how different types combine, not from oscillations within each type.
Over the rationals, the mechanism is different. Conditioning on L-function values creates covariance between Frobenius traces and periods through Euler product constraints. A fixed L-value constrains the product of local factors, which induces correlations between individual factors at different primes. Wachs proves this covariance converges to an explicit function proportional to the inverse square root of prime size.
The empirical finding is striking: the covariance changes sign depending on L-value magnitude. Positive for small L-values, negative for large ones. Analysis of over 650,000 elliptic curves confirms that the covariance concentrates in the Tamagawa product — the ratio of periods — rather than in other arithmetic invariants.
The contribution is identifying the mechanism. Murmurations were first observed as unexplained oscillatory patterns in scatter plots of arithmetic data. The explanation is that Euler product constraints propagate information between local factors at different primes, and this propagation creates the oscillatory structure. The local factors — the most elementary objects in the arithmetic hierarchy — are doing the work. The global pattern emerges from local constraints. The sign change is where the structure lives.