The "movement module". I have a friend with a 44mm Offshore, same caliber as yours (and mine, I have the same Offshore as you on right now as I type!) and it encountered this issue. The watch worked fine for months, and then began to stop running when off the wrist. Here in the US, the AP Service Center in Clearwater, Florida sends a "return kit" to their clients. They FedEx a parcel with full instructions on how to return it, and even include an insured return shipping label. It could not be any easier. I think he said the watch was back on the wrist in less than 3 weeks, with the defective part replaced.
I understand how disappointing it is (believe me, I do, I have needed to utilize factory service for 3 different watches in the past year +). The good news is no brand does it better than Audemars Piguet. You are in good hands and they will make it right.
Please keep us posted.