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On a Problem of Garcia, Stichtenoth, and Thomas

In a recent paper, Garcia, Stichtenoth, and Thomas exhibited, for every finite field E that is not a prime field, an explicit sequence of absolutely irreducible smooth projective curves Cn over E with genus tending to infinity and with #Cn(E)/genus(Cn) tending to a positive limit. I show that their construction does not work over prime fields.