  Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation
are considered in the focusing case. By using one-fold and two-fold Darboux
transformations, we construct explicitly the rogue periodic waves of the mKdV
equation expressed by the Jacobian elliptic functions dn and cn respectively.
The rogue dn-periodic wave describes propagation of an algebraically decaying
soliton over the dn-periodic wave, the latter wave is modulationally stable
with respect to long-wave perturbations. The rogue cn-periodic wave represents
the outcome of the modulation instability of the cn-periodic wave with respect
to long-wave perturbations and serves for the same purpose as the rogue wave of
the nonlinear Schrodinger equation (NLS), where it is expressed by the rational
function. We compute the magnification factor for the cn-periodic wave of the
mKdV equation and show that it remains the same as in the small-amplitude NLS
limit for all amplitudes. As a by-product of our work, we find explicit
expressions for the periodic eigenfunctions of the AKNS spectral problem
associated with the dn- and cn-periodic waves of the mKdV equation.
