It needs to be below the frost depth. I don't know how much frost penetration you get in the arena? Today I thought I was going to level out some well-composted horse manure in a new flower bed. Surprise! It was still frozen hard as a rock about 10" down. But, that is outside.
My best guess is to dig a trench starting at 4 feet deep, put 6" of washed stone in the bottom, then the tile, then another 18" of stone, then filter fabric. That would bring the stone to within 2 feet of the current grade, which you plan to top with 4"-5" of sand. If you can connect the tile with a header at one end of the arena and then run one tile from the header out through a doorway, that would be good.
A slope of 1/4" per foot should be enough. The total drop over 100' would be: 100 x 0.25" = 25". So, if the arena is 100' long, the tile would be 3'-6" + 2'-1" (5'-7") below grade where it leaves the barn. That should be below the frost depth outside the arena. Do you have someplace lower than that within a reasonable distance from the arena to run the tile to daylight at about 1/8" per foot slope?
Depending on the frost depth inside and outside the arena, and how much fall there is outside to drain to, you have to fudge around until you get a fit I guess. You can use a slope less than 1/4" per foot, but no less than 1/8" per foot.
If you run the tile across the short width of the arena, connecting to a header tile--that runs the length of the arena--with tee joints, you will have less total slope depth needed than running it lengthwise.