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.

Figures (7)

• Figure 1: The $L(1)$-adjusted correlation $\operatorname{Corr}(a_p, \Omega \mid L(1))$ decays as $C/\sqrt{p}$ with $C = 0.166$, consistent with the single-prime mechanism of \ref{['thm:C']}.
• Figure 2: The adjusted correlation $\operatorname{Corr}(a_p, \Omega \mid L(1))$ averaged over $p \leq 13$ is flat across five conductor windows, confirming the $o(1)$ convergence of \ref{['thm:C']}.
• Figure 3: The Tamagawa channel $\operatorname{Corr}(a_3, 1/\!\prod c_v)$ by $L(1)$ decile. Blue bars show the Tamagawa channel (red where negative); the orange line shows the full period correlation $\operatorname{Corr}(a_3, \Omega)$. The sign flip at decile 10 confirms the CFKRS prediction $C(c) < 0$ for large $c$.
• Figure 4: CFKRS predicted $C(c)/\!\sqrt{3}$ (dashed, right axis) vs. observed $\operatorname{Corr}(a_3, 1/\!\prod c_v)$ (solid, left axis) by $L(1)$ decile. Both curves decrease from positive to negative, confirming the sign-flip prediction, but the observed non-monotonicity at deciles 7--9 limits the shape correlation to $r = +0.25$.
• Figure 5: Left: $N_\lambda(q)/q$ on a log scale; marker shapes distinguish residue classes mod $96$, revealing the quasi-polynomial branching. Right: type proportions converge to ${\sim}71\%$ ($\Phi_2^2\Phi_6$) and ${\sim}27\%$ ($\Phi_4^2$).

Theorems & Definitions (13)

• Theorem 1: Kronecker obstruction
• Theorem 2: Bessel barrier
• Theorem 3: Period--trace covariance
• proof : Proof that $|\text{\fontencoding{T2A}\CYRSH}|$ is a type invariant
• Remark 4
• Remark 5: Density-weighted positivity
• Remark 6
• Remark 7
• Remark 8
• Remark 9