First off, you should probably get an aviation lawyer. AOPA might be able to help if you have their legal plan (that plan is a good idea for non-airline pilots).
The FAA can deny any class medical for substance abuse issues. They might be wondering if this was alcohol related.
Since an ATP requires the holder to be of "good moral character", I believe they can deny a first class medical based on criminal history.
Also, I suspect the FAA might consider a criminal history to be indication of mental or emotional issues, which ARE grounds for denial per FAR 63.
If you're lucky they will simply determine that this was a one-time only event, and not indicative of your mental state.
Also, if you are planning on working for an airline, be advised that there is a long list of crimes which will disqualify you from working at an airport for ten years. This is a federal law, the airlines have no say, they simply cannot employ you...better check that list and make sure your conviction is not on it.