But in any case, I'd imagine that, like in past games, the crime you commited to warrant your arrest will be left up to the player to decide, though if you're supposed to be scheduled for execution, it seems to narrow it down somewhat, as it can't possibly be a petty crime,
Well, it depends. Our modern society considers murder one of the worst crimes possible. Other, older societies might not agree. Death sentences were handed out rather liberally in Britain during the middle ages, and I think they regarded offences towards property as just as bad as offences against another persons life. Stealing could easily lead you to the gallows. So it depends on the laws in Skyrim (the region, not necessarily nordic - and that's IF you actually have been sentenced, and aren't just being summarily executed without any sentencing.)
But, then again, moral views in TES tend to correspond at least somewhat with todays views in our world - pickpocketing/shoplifting is naughty, armed robbery is bad, and murder is a big no-no. My guess is we either won't be told, or it will be in relation to the ongoing civil war. Probably an unlawful execution.
Or the execution could trigger the choice of sides. Join us, or we'll execute you. Refuse, and the other side rescues you in the nick of time.