Vianney, firstly welcome to TM.
Your starting problem sounds exactly the same as the problem I had with 2009 Aero on the first day I went to drive it. It started and reversed out of the garage OK, when I tried to start it again after locking the garage....... nothing. The ignition worked, lights etc, but absolutely nothing when the starter button was pressed.
It turned out to be a failed soldered joint in the starter circuit somewhere. I am sorry I can't tell you more, the car ran fine with a bump start to get it to my nearest agent, who found and fixed the fault.
I hope you get it sorted soon and get back on the road.
