As short-sleeved shirts are technically allowed, it's your choice if you want to wear one, but I will say that I've never seen a short sleeved shirt at a rated show. (ETA: in the hunter ring. If you're doing the jumpers, then there are lots of cute short sleeve options). Even under a jacket, a short sleeve shirt won't give the same presentation as a long sleeved shirt, as it's typical to see your cuffs poke out beyond your jacket.
I'd recommend looking into one of the newer style shirts, with wrap collars, that are made of extra breathable fabric. The wrap collars are very popular right now and, as a bonus, you don't have to worry about losing a choker! Mine are all very lightweight and comfortable as well, and I live in the southeast, so we have a pretty hot climate!