This is easily one of the hardest achievements to claim unless you feel like waking up around 4am on a sunday morning when the leaderboards are reset. If you feel like doing this, pick a mountain and find any red flagged event. Beat any one of these challenges, so you then have a controlling point on the mountain, then beat the session score. Your achievement should pop up, if it doesn't go back to the main menu and refresh the sponsor. The reason this seems so simple is there are no sponsors when the leaderboard for the mountains are reset. So, once you own one flagged challenge you will have the controlling portion of the mountain's battlegrounds.
Alternatively, you can do this in the week, but it's going to require far more effort on your part. If you are sponsored by "SDS Rome" you will need to own four of the eight flagged challenges on the mountain. This means beating four other battleground scores unless your sponsor already owns two other battlegrounds. If this is the case, you will only need to own two. If you are not sure, drop in at a few different drop points and locate each challenge, if it has a green flag your sponsor already owns that battleground.
If this doesn't work for you, then you probably don't have the controlling half of the mountain. I think there are two flagged battlegrounds on each drop point, which would make 16 challenges, so overall you would be to beat and own eght different flagged events. I recommend beating the mountains session score first and then move onto the flagged battlegrounds.
The leaderboards are reset weekly on Sundays at 4 AM GMT.