  The elastic moduli, elastic anisotropy coefficients, sound velocities and
Poisson's ratio of hcp solid helium have been calculated using density
functional theory in generalized gradient approximation (up to $30$ TPa), and
pair+triple semi-empirical potentials (up to 100 GPa). Zero-point vibrations
have been treated in the Debye approximation assuming $^4$He isotope (we
exclude the quantum-crystal region at very low pressures from consideration).
Both methods give a reasonable agreement with the available experimental data.
Our calculations predict significant elastic anisotropy of helium ($\triangle P
\approx 1.14$, $\triangle S_1 \approx 1.7$, $\triangle S_2 \approx 0.93$ at low
pressures). Under terapascal pressures helium becomes more elastically
isotropic. At the metallization point there is a sharp feature in the elastic
modulus $C_S$, which is the stiffness with respect to the isochoric change of
the $c/a$ ratio. This is connected with the previously obtained sharp minimum
of the $c/a$ ratio at the metallization point.
  Our calculations confirm the previously measured decrease of the Poisson's
ratio with increasing pressure. This is not a quantum effect, as the same sign
of the pressure effect was obtained when we disregarded zero-point vibrations.
At TPa pressures Poisson's ratio reaches the value of $0.31$ at the theoretical
metallization point ($V_{mol}=0.228$ cm$^3$/mol, $p=17.48$ TPa) and $0.29$ at
30 TPa. For $p=0$ we predict a Poisson's ratio of $0.38$ which is in excellent
agreement with the low-$p$-low-$T$ experimental data.
