Pepper plant leaves curl upward for reasons such as not getting enough water and transplant shock, according to SFGate. Viruses are another cause of leaf curl. However, pepper plants with viral issues typically exhibit other symptoms, including spots and rings on the leaves.
Pepper plants grown in excessively dry conditions also can experience upward leaf curl. Typically, the leaves on underwatered or dry pepper plants look yellow and start curling at the bottom of the plant. Overwatering and failure to properly drain the soil around pepper plants also can cause leaf curl. Leaf curl also can be the result of pests such as aphids and mites, according to Gardening Know How.