On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$mathbb {F}_{varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities ( # E( mathbb {F}_{q^k} ) ) and ( # E( mathbb {F}_q ) ) of an elliptic curve E over ( mathbb {F}_q ). We also deduce information about the structure of the 2-Sylow subgroup ( E[ 2^infty ]( mathbb {F}_{q^k} ) ) from the exponents of ( E[ 2^infty ]( mathbb {F}_q ) ).