On the exponent of the group of points of an elliptic curve over a finite field

We present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields $\mathbb{F}_{q^m}$ where either $q$ is fixed or $m=1$ and $q$ is prime. Here, we let both $q$ and $m$ vary; our estimate is explicit and does not depend on the elliptic curve.