  In this paper, we study the luminosity function and formation rate of short
gamma-ray bursts (sGRBs). Firstly, we derive the $E_p-L_p$ correlation using 16
sGRBs with redshift measurements and determine the pseudo redshifts of 284
Fermi sGRBs. Then, we use the Lynden-Bell c$^-$ method to study the luminosity
function and formation rate of sGRBs without any assumptions. A strong
evolution of luminosity $L(z)\propto (1+z)^{4.47}$ is found. After removing
this evolution, the luminosity function is $ \Psi (L) \propto L_0 ^ {- 0.29 \pm
0.01} $ for dim sGRBs and $ \psi (L) \propto L_0 ^ {- 1.07 \pm 0.01} $ for
bright sGRBs, with the break point $8.26 \times 10^{50} $ erg s$^{-1}$. We also
find that the formation rate decreases rapidly at $z<1.0$, which is different
with previous works. The local formation rate of sGRBs is 7.53 events
Gpc$^{-3}$ yr$^{-1}$. Considering the beaming effect, the local formation rate
of sGRBs including off-axis sGRBs is $ 203.31^{+1152.09}_{-135.54} $ events
Gpc$^{-3}$ yr$^{-1}$. We also estimate that the event rate of sGRBs detected by
the advanced LIGO and Virgo is $0.85^{+4.82}_{-0.56} $ events yr$^{-1}$ for
NS-NS binary.
