  We have undertaken a project to investigate the host galaxies and
environments of a sample of quasars at z~4. In this paper, we describe deep
near-infrared imaging of 34 targets using the Magellan I and Gemini North
telescopes. We discuss in detail special challenges of distortion and
nonlinearity that must be addressed when performing PSF subtraction with data
from these telescopes and their IR cameras, especially in very good seeing. We
derive black hole masses from emission-line spectroscopy, and we calculate
accretion rates from our K_s-band photometry, which directly samples the
rest-frame B for these objects. We introduce a new isophotal diameter technique
for estimating host galaxy luminosities. We report the detection of four host
galaxies on our deepest, sharpest images, and present upper limits for the
others. We find that if host galaxies passively evolve such that they brighten
by 2 magnitudes or more in the rest-frame B band between the present and z=4,
then high-z hosts are less massive at a given black hole mass than are their
low-z counterparts. We argue that the most massive hosts plateau at <~10L*. We
estimate the importance of selection effects on this survey and the subsequent
limitations of our conclusions. These results are in broad agreement with
recent semi-analytical models for the formation of luminous quasars and their
host spheroids by mergers of gas-rich galaxies, with significant dissipation,
and self-regulation of black hole growth and star-formation by the burst of
merger-induced quasar activity.
