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Jul 21, 2018 at 7:39 comment added Bombyx mori @WillSawin: Thanks! I am really happy to know.
Jul 21, 2018 at 7:38 comment added Will Sawin One can do this for the etale fundamental group. This requires the fact that the curve is projective (or a characteristic zero assumption) to ensure $\pi_1(X \times X) \cong \pi_1(X) \times \pi_1(X)$, it is false for some groups in characteristic $p$.
Jul 21, 2018 at 7:29 history answered Bombyx mori CC BY-SA 4.0