Thank you Richard for bringing up this topic. It is a very interesting subject. Maybe it would be a good beginning to show (again) the few passages from the NT where this expression is used.
The expression "hupo nomos" is a greek idiom which is translated "under the law", and is used by Paul several times and is found in the books of Romans, Corintians and Galatians:
Rom 6:14.15 - For sin shall not have dominion over you: for ye are not under the law, but under grace. What then? shall we sin, because we are not under the law, but under grace? God forbid.
1Co 9:20.21 - And unto the Jews I became as a Jew, that I might gain the Jews; to them that are under the law, as under the law, that I might gain them that are under the law; To them that are without law, as without law, (being not without law to God, but under the law to Christ,) that I might gain them that are without law.
Gal 3:23-27 - But before faith came, we were kept under the law, shut up unto the faith which should afterwards be revealed. Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. But after that faith is come, we are no longer under a schoolmaster. For ye are all the children of God by faith in Christ Jesus. For as many of you as have been baptized into Christ have put on Christ.
Gal 4:4-6 - But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law, To redeem them that were under the law, that we might receive the adoption of sons. And because ye are sons, God hath sent forth the Spirit of his Son into your hearts, crying, Abba, Father.
Gal 4:21 - Tell me, ye that desire to be under the law, do ye not hear the law?
Gal 5:18 But if ye be led of the Spirit, ye are not under the law.