Strong Weil curves over F_q(T) with small conductor

Details Strong Weil curves over F_q(T) with small conductor Strong Weil curves over F_q(T) with small conductor

2010-02-11

arxiv.org/abs/1002.2260v2

J. Number Theory 131 (2011), 285-299

Mathematics

Number Theory

published version, minor changes, new address, 20 pages, contains standard LaTeX-graphics

Scientific paper

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 $\bar{{\mathbb F}_q}$.

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