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Let C be a geometrically connected and geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that

• $L \cap K' = K$, and
• $C(L) \neq \emptyset$?

Note that the hypotheses on C are necessary -- the curve x^2 + y^2 = 0, with the origin removed, is not geometrically integraland geometrically connected, but gives a counterexample for K = Q and K' = Q(i).

Also, I can prove that this is true when C has prime gonality. It would be odd, though, for this to be a necessary hypothesis.

2 typo, added a short remark

Let C be a geometrically connected and geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that

• $L \cap K' = K$, and
• $C(L) \neq \emptyset$?

Note that the hypotheses on C is are necessary -- the curve x^2 + y^2 = 0, with the origin removed, is not geometrically integral and geometrically connected, but gives a counterexample for K = Q and K' = Q(i).

Also, I can prove that this is true when C has prime gonality. It would be odd, though, for this to be a necessary hypothesis.

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# Existence of points on varieties which avoid a given number field.

Let C be a geometrically connected and geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that

• $L \cap K' = K$, and
• $C(L) \neq \emptyset$?

Note that the hypotheses on C is necessary -- the curve x^2 + y^2 = 0, with the origin removed, is not geometrically integral and geometrically connected, but gives a counterexample for K = Q and K' = Q(i).