Let $E$ be an elliptic curve over $\mathbb{C} (t)$ (rational functions). I require $E$ to be defined by the following equation.

$$ y^2 = x^3 + A x + B$$

Where $A, B \in \mathbb{C} (t)$.

Question: Is it possible that there is no $\mathbb{C} (t)$ points on E.

Comment: To make this question more elementary. Is there such $x, y \in \mathbb{C} (t)$ which are solutions of the equation.


Yes, this is possible. In general, your curve $E/\mathbb{C}(t)$ satisfies the Mordell-Weil Theorem (see Chapter III, Theorem 6.1, of "Advanced Topics in the Theory of Elliptic Curves" by Silverman), and the Mordell-Weil group can be trivial, so there would be no points.

For instance, in this article, Cox shows that the curve $y^2=4x^3-3x-t$ has trivial Mordell-Weil group over $\mathbb{C}(t)$.


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