Murmurations, Periods, and Local Factors

Abstract

We prove that over function fields F_q(t), the Tate-Shafarevich group |Sha| is an invariant of the cyclotomic type of the L-polynomial, so that |Sha|-stratified murmuration densities reduce to type-weighted densities with no within-type zero displacement (Theorem A). Over Q, the obstruction vanishes because Satake parameters are continuous: conditioning on L(f,1) = c biases each theta_p through the Euler product constraint, creating covariance between the Frobenius trace a_p and the real period Omega_f that the L-value regression does not absorb. A new inequality for modified Bessel functions (Theorem B) establishes the positivity of this single-prime covariance under a linearized tilt; the full Euler-factor positivity follows by perturbation for large p (Theorem 10) and is verified numerically for small primes (Theorem 11). We establish that the conditional covariance Cov(a_p, Omega_f | L(f,1) ~ c, N) converges to an explicit function C(c)/sqrt(p) as N -> infinity (Theorem C). The function C(c) changes sign -- positive at small c, negative at large c -- a prediction confirmed empirically using 657,000 curves from the Cremona database. Empirically, the covariance is concentrated entirely in the Tamagawa product prod c_v: at fine L(1)-conditioning, the Tamagawa channel accounts for 100% of the signal, and cross-validated regression confirms that the nonlinear adjoint-Euler-factor weighting carries independent Tamagawa information beyond a full trace basis but adds nothing for |Sha|. The |Sha|-modulation of murmurations discovered in [Wac26a] is a consequence of the BSD identity linking |Sha| to local factors -- the same local-factor mechanism operates discretely over function fields and continuously over Q.