Surjectivity of mod 2ⁿ representations of elliptic curves

الأصل · EN

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL₂(Z/lⁿ Z). There are three exceptional prime powers lⁿ=2,3,4 when surjectivity of the mod lⁿ representation does not imply that for l(n+1). Elliptic curves with surjective mod 3 but not mod 9 representation have been classified by Elkies. The purpose of this note is to do this in the other two cases.