There could be a number of reasons for pain under the armpit.
In the same way as the blood vessels of the heart can cause chest pain and heart attacks, narrowing of the major blood vessels supplying the arm (peripheral artery disease) can lead to pain that is felt in the armpit.
Also, Angina is a condition that does cause severe chest pain due to a temporary reduction in blood flow to the heart. The pain occurs because the heart muscle does not receive enough oxygenated blood to keep up with its activity. Symptoms of this condition can cause pain under your left armpit and shoulder.
Fatigue can also cause pain under the armpit.