We present the analysis of Herschel SPIRE far-infrared (FIR) observations of the z = 2.515 lensed galaxy SMM J163554.2+661225. Combining new 250, 350, and 500 micron observations with existing data, we make an improved fit to the FIR spectral energy
distribution (SED) of this galaxy. We find a total infrared (IR) luminosity of L(8--1000 micron) = 6.9 +/- 0.6x10^11 Lsol; a factor of 3 more precise over previous L_IR estimates for this galaxy, and one of the most accurate measurements for any galaxy at these redshifts. This FIR luminosity implies an unlensed star formation rate (SFR) for this galaxy of 119 +/- 10 Msol per yr, which is a factor of 1.9 +/- 0.35 lower than the SFR derived from the nebular Pa-alpha emission line (a 2.5-sigma discrepancy). Both SFR indicators assume identical Salpeter initial mass functions (IMF) with slope Gamma=2.35 over a mass range of 0.1 - 100 Msol, thus this discrepancy suggests that more ionizing photons may be necessary to account for the higher Pa-alpha-derived SFR. We examine a number of scenarios and find that the observations can be explained with a varying star formation history (SFH) due to an increasing star formation rate (SFR), paired with a slight flattening of the IMF. If the SFR is constant in time, then larger changes need to be made to the IMF by either increasing the upper-mass cutoff to ~ 200 Msol, or a flattening of the IMF slope to 1.9 +/- 0.15, or a combination of the two. These scenarios result in up to double the number of stars with masses above 20 Msol, which produce the requisite increase in ionizing photons over a Salpeter IMF with a constant SFH.
