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Meromorphic functions on $U^2 = T^3 + 1$, cokernel of $O_S \to F_\infty/O_\infty$ [closed]

See here. Crossposted from math.stackexchange since there's no good answer despite $>$ 20 upvotes. Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ ge …