On the torsion of optimal elliptic curves over function fields

For an optimal elliptic curve $E$ over $\F_q(t)$ of conductor $\fp\cdot\infty$, where $\fp$ is prime, we show that $E(F)_\tor$ is generated by the image of the cuspidal divisor group. We also show that $E(F)_\tor\cong \Z/n\Z$ for some $n$, $1 \leq n\leq 3$, and that $n$ divides $(q-1)$ and $\deg(\fp)$.

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Published 1 January 2006