  The recent discovery of bulk superconductivity in trilayer nickelate
La$_4$Ni$_3$O$_{10}$ with the critical temperature $T_c$ near $30$K under high
pressure is attracting a new wave of research interest, after the breakthrough
of bilayer La$_3$Ni$_2$O$_7$ with $T_c$ near $80$K. The similarities and
differences of electronic structure and superconducting mechanism in these two
systems are urgent theoretical issues. In this Letter, we study the electronic
band structure and construct a minimal trilayer tight-binding model for the
high-pressure phase of La$_4$Ni$_3$O$_{10}$ in terms of the nickel
$3d_{x^2-y^2}$ and $3d_{3z^2-r^2}$ orbitals, and study the superconducting
mechanism due to local Coulomb interactions by the unbiased functional
renormalization group. We find antiferromagnetic correlations between the outer
layers instead of neighboring ones, apart from the inplane correlations. The
effective interaction induces Cooper pairing with the $s_\pm$-wave symmetry,
which changes sign across the Fermi pockets. We find $T_c$ in
La$_4$Ni$_3$O$_{10}$ is systematically lower than that in La$_3$Ni$_2$O$_7$,
and electron doping can enhance $T_c$.
