  We present a new cosmological galaxy formation model, $\nu^2$GC, as an
updated version of our previous model $\nu$GC. We adopt the so-called
"semi-analytic" approach, in which the formation history of dark matter halos
is computed by ${\it N}$-body simulations, while the baryon physics such as gas
cooling, star formation and supernova feedback are simply modeled by
phenomenological equations. Major updates of the model are as follows: (1) the
merger trees of dark matter halos are constructed in state-of-the-art ${\it
N}$-body simulations, (2) we introduce the formation and evolution process of
supermassive black holes and the suppression of gas cooling due to active
galactic nucleus (AGN) activity, (3) we include heating of the intergalactic
gas by the cosmic UV background, and (4) we tune some free parameters related
to the astrophysical processes using a Markov chain Monte Carlo method. Our
${\it N}$-body simulations of dark matter halos have unprecedented box size and
mass resolution (the largest simulation contains 550 billion particles in a
1.12 Gpc/h box), enabling the study of much smaller and rarer objects. The
model was tuned to fit the luminosity functions of local galaxies and mass
function of neutral hydrogen. Local observations, such as the Tully-Fisher
relation, size-magnitude relation of spiral galaxies and scaling relation
between the bulge mass and black hole mass were well reproduced by the model.
Moreover, the model also well reproduced the cosmic star formation history and
the redshift evolution of rest-frame ${\it K}$-band luminosity functions. The
numerical catalog of the simulated galaxies and AGNs is publicly available on
the web.
