In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

In this paper we investigate Hesse’s elliptic curves $H_{\mu }:U^3+V^3+W^3=3\mu UVW,\mu \in \mathbf{Q}-\left\{1\right\}$, and construct their twists, $H_{\mu ,t}$ over quadratic fields, and $\tilde{H}(\mu ,t),\mu ,t\in \mathbf{Q}$ over the Galois closures of cubic fields. We also show that $H_{\mu }$ is a twist of $\tilde{H}(\mu ,t)$ over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, $R(t;X):=X^3+tX+t,t\in \mathbf{Q}-\{0,-27/4\}$, to parametrize all of quadratic fields and cubic ones. It should be noted that $\tilde{H}(\mu ,t)$ is a twist of $H_{\mu }$ as algebraic curves because it may not always have any rational points...