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Mar 7, 2018 at 13:58 comment added Lubin I’m both too lazy and too groggy to check, but I believe you can get a counterexample to the converse if you look in genus three.
Mar 7, 2018 at 8:41 vote accept Ofir Gorodetsky
Mar 6, 2018 at 22:50 comment added skd Regarding your last line: as I'm sure you know, the converse is true for supersingular elliptic curves.
Mar 6, 2018 at 22:45 history answered Felipe Voloch CC BY-SA 3.0