Strong Weil curves over F_q(T) with small conductor

Abstract

We continue work of Gekeler and others on elliptic curves over ${\mathbb F}_q(T)$ with conductor $\infty\cdot{\mathfrak n}$ where ${\mathfrak n}\in{\mathbb F}_q[T]$ has degree 3. Because of the Frobenius isogeny there are infinitely many curves in each isogeny class, and we discuss in particular which of these curves is the strong Weil curve with respect to the uniformization by the Drinfeld modular curve $X_0({\mathfrak n})$. As a corollary we obtain that the strong Weil curve $E/{\mathbb F}_q(T)$ always gives a rational elliptic surface over $\bar{{\mathbb F}_q}$.