  We prove an upper bound for the ground state energy of a Bose gas consisting
of $N$ hard spheres with radius $\mathfrak{a}/N$, moving in the
three-dimensional unit torus $\Lambda$. Our estimate captures the correct
asymptotics of the ground state energy, up to errors that vanish in the limit
$N \to \infty$. The proof is based on the construction of an appropriate trial
state, given by the product of a Jastrow factor (describing two-particle
correlations on short scales) and of a wave function constructed through a
(generalized) Bogoliubov transformation, generating orthogonal excitations of
the Bose-Einstein condensate and describing correlations on large scales.
