I've been having the 31/37 issue and recently found out yesterday what the real problem was. I will mention the solution a little later in my post. I'm a small car dealer and bought one of these PT's (03 GT 2.4 Turbo) with the 31 & 37 issue. The internet has been invaluable to me in fixing issues in any car that I buy, and believe me, they all have them. So I thought I'd give back a little and share with everyone how to fix this problem. The 31 & 37 issue is very common with these cars, but all too often, someone posts a problem, ends up taking it to the dealer, then never bothers to take the time to share the solution. The internet is filled with dead-end threads regarding the 31/37 issue. This one is by far the best, but still incomplete. I hope to put this issue to bed once and for all.
While following this thread, I did what George W did and ordered a new PCM from ACR. Installed it, cleared the codes, reset the monitors by driving, and was able to run it through emissions testing. I figured this was the end of it. Well, while driving home one day, the codes came back. They were stubborn as hell, my Innova scanner wasn't able to clear them. All that it could do was reset the monitors. The codes were also both stored & pending and I couldn't reset the monitors no matter how far I drove the car. I took it to a small indy shop and they were able to do a 'relearn' on the system, which did clear the light and I was able to reset the monitors, but the light came back after about 100 miles. In between repairs, I pulled my hair out checking grounds, heater circuit resistance, wiring harnesses. The 02 sensors were new, probably replaced by the previous owner in an attempt to fix the problem. There is a coolant line that runs behind the engine that had leaked some coolant on the wiring harness below which I thought could be a problem. I thought I was getting somewhere, but once I took the insulation off, it turned out the wires were fine. I finally bit the bullet and took it to a dealer for a solution.
Diagnosis---PCM drivers defective
The solution is to replace the PCM (unless you have other obvious problems like bad 02 sensors/grounds/etc). My original diagnosis (based on George W's experience) was correct. However, at this point I'm not overly confident that ACR can fix it. They're sending me a replacement PCM, but for all I know, it could be the old core that I sent them and I feel therein lies the problem. This issue seems to pop up more as these cars age. 11 years isn't too terribly old for today's modern cars, but it is for the electrical components. If you've noticed, most PCM's have heat sinks on them. Even though they're usually not directly placed by a heat source in the engine bay, they still generate a lot of heat through electrical resistance. I feel that this is the real issue. Whoever gets my old PCM will be getting an 11 year old component subjected to 11 years of heat and wear and tear. This combined with what I feel is sketchy electrical design/software by Chrysler and you have a recipe for disaster. This is why I'm not sure that the replacement unit will cure the problem any better than the first. The only other solution is to go with a new factory PCM from the dealer, but that could run $400-$600 depending on your location.
I hope this helps anyone out there having the same issue.