  We present new results on the cosmic star formation history in the SXDS-UDS
field out to z=1.6. We compile narrow-band data from the Subaru Telescope and
the Visible and Infrared Survey Telescope for Astronomy (VISTA) in conjunction
with broad-band data from the SXDS and UDS, to make a selection of 5725
emission-line galaxies in 12 redshift slices, spanning 10 Gyr of cosmic time.
We determine photometric redshifts for the sample using 11-band photometry, and
use a spectroscopically confirmed subset to fine tune the resultant redshift
distribution. We use the maximum-likelihood technique to determine luminosity
functions in each redshift slice and model the selection effects inherent in
any narrow-band selection statistically, to obviate the retrospective
corrections ordinarily required. The deep narrow-band data are sensitive to
very low star formation rates (SFRs), and allow an accurate evaluation of the
faint end slope of the Schechter function, alpha. We find that alpha is
particularly sensitive to the assumed faintest broadband magnitude of a galaxy
capable of hosting an emission line, and propose that this limit should be
empirically motivated. For this analysis we base our threshold on the limiting
observed equivalent widths of emission lines in the local Universe. We compute
the characteristic SFR of galaxies in each redshift slice, and the integrated
SFR density, rho_{SFR}. We find our results to be in good agreement with the
literature and parametrize the evolution of the SFR density as rho_{SFR}
proportional to (1 + z)^{4.58} confirming a steep decline in star formation
activity since z = 1.6.
