Strong Weil curves over F_q(T) with small conductor

TL;DR

This paper investigates strong Weil elliptic curves over function fields with small conductors, identifying the unique strong Weil curve in each isogeny class and its implications for rational elliptic surfaces.

Contribution

It characterizes the strong Weil curves over ${f F}_q(T)$ with degree 3 conductor and explores their properties related to Frobenius isogeny and modular uniformization.

Findings

Existence of a unique strong Weil curve in each isogeny class.

Strong Weil curves correspond to rational elliptic surfaces.

Explicit description of curves with degree 3 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 $\bar{{\mathbb F}_q}$.