I hope I fixed it!
I had to make a “special cable” to connect the FXT003 to the PC. It has the same wiring as the 15-pin <-> 9 pin cable delivered with the FXT003 box PLUS a 10 kohm resistor between the pin Nr3 of the 15-pin connector (BOOT) and the ground.
With this cable, dwlwin works (first launch dwlwin, then switch the power on the modem).
With this, I can download the bootloader and the firmware (7.44 version).
At this point,the AT commands work fine.
And after that with Developer Studion, I can begin to manage my project.
Just now, the “LUA interactive shell” works.
This has been done thanks to this forum and his contributors. The Sierra and Wavecom documentation is difficult to use. As we say in French:
“Why to make it simple when it is so easy to make it complicated”.