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It looks like it is a problem with the anit-theft system. Code p1610 is for an Immobilizer System Malfunction, usually set because the wrong key code was picked up by the anti-theft system. The procedure for checking it says to use a spare key that has worked in the past and if that doesn't fix it, you'll either have to have a licensed locksmith with the ability to reprogram your car to your key come out, or take it into the dealership to have it reprogrammed.
The most likely cause for this is that haveing no battery power for a while as you described allowed the programmed key information to be lost.