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Sep 7, 2020 at 10:14 vote accept ChrisLazda
Sep 7, 2020 at 10:14 comment added ChrisLazda That's great thanks!
Sep 5, 2020 at 11:23 history edited Jérôme Poineau CC BY-SA 4.0
Answer rewritten since I had missed the point before.
Sep 5, 2020 at 11:17 comment added Jérôme Poineau I was indeed answering something completely different. I will give it another try.
Sep 5, 2020 at 7:54 comment added ChrisLazda Hi Jerome, I was actually about to email you this question, so I'm glad you've popped up here! I think I perhaps didn't explain it very well - the question was more about whether $X_L$ can ever be a topological covering space of $X$ - i.e. a covering space map on the underlying topological spaces. It boils down to the question of whether or not every point of $X$ (of any Type) has precisely $[L:k]$ preimages in $X_L$. It feels like this is unlikely - for conics, I feel as though it should be possible to cook up some Type II point with only one preimage, but I didn't manage to do so.
Sep 4, 2020 at 19:25 history answered Jérôme Poineau CC BY-SA 4.0