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My name is XXXXX XXXXX I will help you with this -
Ok- you have the F1CE0481D engine in your vehicle - and given the symptoms you outline, this could be as simple as a crank sensor or Ect ( engine temp sensor ) fault. Both are common issues / faults on this engine....without fault codes, its hard to say exactly which is the issue if any at the moment.
As the injection light (fault) light comes on, that will mean the Ecu has logged fault codes - but - you need to be testing for the fault codes , while the light is on. Once its off, the codes are gone.
Other issues that you can have that can and will cause such faults are faults with the fuel pump or one or more of the injectors. If one injector for example is faulty it causes a pressure drop in the system so the Ecu wont let the engine start.
You could try changing the ECT sensor , they are not expensive - You would need to get one from a local Fiat dealer parts dept - and they will tell you where it is on the engine if you are not sure. It would be a good move at present as mentioned, its a common issue and that could just be it.
If not though , then I have to strongly advise to have either a garage with the proper equipment on it to test it or, better still a dealer or a diesel specialist - who has the correct equipment to check the live data - fuel system / pressure and so on. There is only so much you can do on your own with these modern systems as you probably know, and the above as I have mentioned is about it...
So work through as above and one way or the other , you will get this sorted out for sure.
Im very pleased to have helped you - Let me know if I can help anymore...
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Kind Rgds - MIKE