1990 miata won't start
 

I have a 1990 Miata with 60,000 miles. It was running perfectly. One day the aftermarket alarm LCD light stayed on after the car was turned off. The next day the alarm worked fine. I live in Ohio and park the Miata in winter. I went to start the car it would'nt start, but I could hear the alarm. I disconnected the alarm andthought the battery was dead. Got a new battery nothing. Can anyone give me a clue? I called my mechanic, he said it sounds electrical. He told me to take the battery out and have it towed to him in the spring.