  We observed nine primary transits of the hot Jupiter TrES-3b in several
optical and near-UV photometric bands from 2009 June to 2012 April in an
attempt to detect its magnetic field. Vidotto, Jardine and Helling suggest that
the magnetic field of TrES-3b can be constrained if its near-UV light curve
shows an early ingress compared to its optical light curve, while its egress
remains unaffected. Predicted magnetic field strengths of Jupiter-like planets
should range between 8 G and 30 G. Using these magnetic field values and an
assumed B_star of 100 G, the Vidotto et al. method predicts a timing difference
of 5-11 min. We did not detect an early ingress in our three nights of near-UV
observations, despite an average cadence of 68 s and an average photometric
precision of 3.7 mmag. However, we determined an upper limit of TrES-3b's
magnetic field strength to range between 0.013 and 1.3 G (for a 1-100 G
magnetic field strength range for the host star, TrES-3) using a timing
difference of 138 s derived from the Nyquist-Shannon sampling theorem. To
verify our results of an abnormally small magnetic field strength for TrES-3b
and to further constrain the techniques of Vidotto et al., we propose future
observations of TrES-3b with other platforms capable of achieving a shorter
near-UV cadence. We also present a refinement of the physical parameters of
TrES-3b, an updated ephemeris and its first published near-UV light curve. We
find that the near-UV planetary radius of Rp = 1.386+0.248-0.144 RJup is
consistent with the planet's optical radius.
