To Sir, with love

15-year-old schoolgirl runs off to France (willingly as it appears) with her 30-year-old married maths teacher.  Has any offence been committed?  The police clearly think so, as they have issued an international arrest warrant against the teacher, and in fact he has now been arrested in France under its authority.  So why?  Well the warrant cites "child abduction".  OK, as she is 15, the girl is legally a minor, but abduction?  It seems to be accepted that she went with him willingly, so how can there be any question of abduction?  And when they get back to this country, what then?  They will presumably want to continue seeing each other - is the law going to try and stop them?  Unless there are things we are not being told, the authorities seem to have chosen a very heavy-handed approach which I feel is only going to make a difficult situation worse.