The magnesium alloys, that have high specific strength, are often applied to the industrial products. However, the magnesium alloys exhibit low ductility at the room temperature on account of its hexagonal close-packed structure. It is difficult to give large deformation to the magnesium alloys at room temperature. Therefore, the plastic forming of a magnesium alloy sheet needs the process at warm temperature. In the present work, the procedure of thermal-mechanical coupled analyses are employed. The numerical simulations of warm deep drawings have been performed in order to evaluate the dependence of the temperature on the plastic forming of a magnesium alloy AZ31 sheet. The mechanical properties of the magnesium alloy AZ31 shall be described as the functions depending on temperature. The shapes of punches and die holes are rectangle whose aspect ratios are 1.5 or 2.0. The corners of punches and dies are heated locally at 473K. The influence of local heating on the formability have been investigated. The relation between the blank size and the formability has been also estimated. As the results of numerical simulations, it was shown that the formability of deep drawing was improved by local heating to the punch and the die. When the blanks of various sizes were tried, the distributions of the plastic strain rate around the die corner were changed. Therefore, the deviation of the plastic flow and the temperature distribution arose in a sheet. Consequently, it is necessary to optimize the blank sizes according to the shape of die holes in addition to the forming temperature.