  The detection of small planets orbiting nearby stars is an important step
towards the identification of Earth twins. In previous work using the Spitzer
Space Telescope, we found evidence to support at least one sub-Earth-sized
exoplanet orbiting the nearby mid-M dwarf star GJ 436. As a follow up, here we
used the Hubble Space Telescope to investigate the existence of one of these
candidate planets, UCF-1.01, by searching for two transit signals as it passed
in front of its host star. Interpretation of the data hinges critically on
correctly modeling and removing the WFC3 instrument systematics from the light
curves. Building on previous HST work, we demonstrate that WFC3 analyses need
to explore the use of a quadratic function to fit a visit-long time-dependent
systematic. This is important for establishing absolute transit and eclipse
depths in the white light curves of all transiting systems. The work presented
here exemplifies this point by putatively detecting the primary transit of
UCF-1.01 with the use of a linear trend. However, using a quadratic trend, we
achieve a better fit to the white light curves and a reduced transit depth that
is inconsistent with previous Spitzer measurements. Furthermore, quadratic
trends with or without a transit model component produce comparable fits to the
available data. Using extant WFC3 transit light curves for GJ436b, we further
validate the quadratic model component by achieving photon-limited model fit
residuals and consistent transit depths over multiple epochs. We conclude that,
when we fit for a quadratic trend, our new data contradict the prediction of a
sub-Earth-sized planet orbiting GJ 436 with the size, period, and ephemeris
posited from the Spitzer data by a margin of 3.1{\sigma}.
