  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.
