>How do I get my horse to not trot or run down the hills?
It sounds like you are starting at the top of the hill. Start at the bottom of the hill, at the walk go up just a few strides, turn and come back down (still walking). Repeat.
Do all of this on a loose rein.
Increase how far you go in small increments as he walks nicely down the shorter segments. For the first day you might just focus on walking up and back on the lowest section of the hill 3 or 4 times and quit there, to quit before you get to the “tough” bits and establish a good theme to hill work. The next day you can work on going further.
If you get to a point where you turn him back down the hill and he doesn’t stay at the walk, turn him back up the hill again, stop, walk a few strides up, then turn and walk a few strides down (not all the way down) then turn and walk a few strides up, then turn and walk a few strides down (a few more than you went up). If you can feel him “start to break” from the walk then turn him BEFORE he breaks.
Remember, you do all of this on a loose rein. It’s his job to stay in the walk without further cues, when you have put him in the walk. You only use the rein to turn him. Pretty soon, when he feels like he’s about to break out of the walk and you lift one rein to turn him he will shift back into a solid walk *without* you needing to actually turn him. Then that single rein lift also becomes a “reminder” cue you can use to help him stay at the walk, but if it becomes repeatedly necessary then you have to turn him and head uphill again.
This also works for barn sour horses. Rushing downhill and rushing back to the barn are the same problem – different triggers (heading downhill, heading back to the barn) but the underlying problem is the same.