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I posted this question on math.SE and it has not been answered. I thought I should move it to MO. But I read this and I decided to follow Asaf Karagila's suggestion, so I am posting my question here. There is a theorem by Poincaré and Hurwitz which says, basically, that an elliptic $\mathbb{Q}$-curve with positive rank has infinitely many points in every ...

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