  We analyze 26 archival Kepler transits of the exo-Neptune HAT-P-11b,
supplemented by ground-based transits observed in the blue (B-band) and near-IR
(J-band). Both the planet and host star are smaller than previously believed;
our analysis yields Rp=4.31 +/-0.06 Earth-radii, and Rs = 0.683 +/-0.009 solar
radii, both about 3-sigma smaller than the discovery values. Our ground-based
transit data at wavelengths bracketing the Kepler bandpass serve to check the
wavelength dependence of stellar limb darkening, and the J-band transit
provides a precise and independent constraint on the transit duration. Both the
limb darkening and transit duration from our ground-based data are consistent
with the new Kepler values for the system parameters. Our smaller radius for
the planet implies that its gaseous envelope can be less extensive than
previously believed, being very similar to the H-He envelope of GJ436b and
Kepler-4b. HAT-P-11 is an active star, and signatures of star spot crossings
are ubiquitous in the Kepler transit data. We develop and apply a methodology
to correct the planetary radius for the presence of both crossed and uncrossed
star spots. Star spot crossings are concentrated at phases -0.002 and +0.006.
This is consistent with inferences from Rossiter-McLaughlin measurements that
the planet transits nearly perpendicular to the stellar equator. We identify
the dominant phases of star spot crossings with active latitudes on the star,
and we infer that the stellar rotational pole is inclined at about 12 +/-5
degrees to the plane of the sky. We point out that precise transit measurements
over long durations could in principle allow us to construct a stellar
Butterfly diagram, to probe the cyclic evolution of magnetic activity on this
active K-dwarf star.
