What can cause sharp pains in my left forearm?

Number of things. Description of sharp makes it likely it is something localized to the forearm and not being 'referred' from another area, like the heart. (see, prior answers related to referred pain). Presuming there's no trauma, insect bite to area, most likely a tendinitis (tendon connects muscle to joint)--rest the muscles, ice, anti inflammatory. Less likely is a clot in the vein. Persistent pain-again, c doc.