What might cause chest pain and left arm numbness?

Rule out heart first: Number one cause in seriousness is heart disease / attack. If you haven't checked this out, do so right away. Other causes include pinched nerve in neck, shoulder injury, overuse syndrome of left arm, thoracic outlet syndrome, blockage of axillary artery.

Hmmm...: Chest pain with exertion radiating to the left arm with numbness or pain, improving with rest is suspicious for heart disease. That needs to be investigated. Alternatively, a pinched nerve in your neck follows that dermatome and can mimic heart disease. Alternatively, arthritis in the shoulder, biceps tendon, or costochondritis all can do this. The key is the last 4 items are non-exertional.