dependng on what it is made of.... I have used a slightly warm iron before. I have also used heavy object (books) to put on it for a while.... any box store sells carpet corner tape also. That works great!
Try two tape or a few tacks.
Usually weighting the corner down with a heavy object will work to reduce the curling.
Buy some carpet tape at a big box store. Measure your area so you get enough tape. Pull up carpet and mat. Lay strips of carpet tape under your carpet. Put carpet back down. You might need to put a new metal carpet edge in so it will lie flat. These are not that expensive. You might need to replace the carpet tack strip by your wall. I’m just not sure where the curled up piece is exactly.
Two sided tape should do it. Put it under the carpet and stand on it.
Curl in in the opposite direction for a few days to let the fibers relax.