A new record for the canonical height on an elliptic curve over \(\mathbb C(t)\) (Q634455)

Let \(E\) be an elliptic curve over \(\mathbb{C}(t)\). The ratio of the degree of the discriminant of \(E\) and the conductor of \(E\) is called the Szpiro ratio of \(E\) and denoted by \(\sigma\). Let \(E\) be a nonconstant elliptic curve over \(\mathbb{C}(t)\) with Szpiro ratio \(\sigma\leq 4\) and \(P\) a nontorsion point in \(E(\mathbb{C}(t))\). The main theorem of this paper asserts that the canonical height \(\hat{h}(P)\) is always greater than \(2987/120120\). Furthermore, the author exhibits the explicit equation for an elliptic curve \(E/\mathbb{C}(t)\) of discriminant degree \(84\) with the Szpiro ratio \(\sigma>4\) having a rational point \(P\) of canonical height \(2987/120120\).