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Murmurations, Periods, and Local Factors
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 $
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