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The automorphism group of an elliptic curve over an algebraically closedfield is well known. However, for various applications in coding theory andcryptography, we usually need to apply automorphisms defined over a finitefield. Although we believe that the automorphism group of an elliptic curveover a finite field is well known in the community, we could not find this inthe literature. Nevertheless, in this paper we show the group structure of theautomorphism group of an elliptic curve over a finite field. More importantly,we characterize subgroups and abelian subgroups of the automorphism group of anelliptic curve over a finite field.  Despite of theoretical interest on this topic, our research is largelymotivated by constructions of optimal locally repairable codes. The firstresearch to make use of automorphism group of function fields to constructoptimal locally repairable codes was given in a paper \cite{JMX20} whereautomorphism group of a projective line was employed. The idea was furthergenerated to an elliptic curve in \cite{MX19} where only automorphisms fixingthe point at infinity were used. Because there are at most $24$ automorphismsof an elliptic curve fixing the point at infinity, the locality of optimallocally repairable codes from this construction is upper bounded by $23$. Oneof the main motivation to study subgroups and abelian subgroups of theautomorphism group of an elliptic curve over a finite field is to remove theconstraints on locality.

The group structures of automorphism groups of elliptic function fields over finite fields and their applications to optimal locally repairable codes