Murmurations, Periods, and Local Factors

Abstract

This note replaces arXiv:2603.22807v1 and withdraws its three theorems, which do not stand as stated. Theorem A, that the order of the Tate-Shafarevich group of $y^2=x^3+x+D(t)$ over ${\mathbb{F}}_q(t)$ is determined by the cyclotomic type of the $L$-polynomial, holds at rank $0$ when $D$ has degree $5$ but fails at positive rank; we give a counterexample and cite the correct statement. Theorem B is Turán's inequality for modified Bessel functions; we prove a version for general even weights and a class of increasing tilts that includes the Euler factor, which gives the Euler-factor tilt at every prime. Theorem C, a limit law for the covariance of $a_p$ with the real period at fixed $L(f,1)$, was not proved, and we explain why. We also recompute the empirical section of the first version from Cremona's tables, correct its numbers, withdraw those we could not reconstruct, and show that its sign flip at $p=3$ and its "100% Tamagawa channel" are artefacts of decile conditioning and of the rank-$0$ BSD formula.

arXiv commentv2 withdraws the three theorems of v1 and corrects its data; 8 pages