[en] We focus on the algebraic area probability distribution of planar random walks on a square lattice with $m_1,m_2,l_1$ and l₂ steps right, left, up and down. We aim, in particular, at the algebraic area generating function $Z_{m_1,m_2,l_1,l_2}(\mathrm{Q})$ evaluated at $\mathrm{Q}={\mathrm{e}}^{{\displaystyle \frac{2\mathrm{i}\mathrm{\pi <!--\; ??\; -->}}{q}}},$ a root of unity, when both $m_1-<!--\; ???\; -->m_2$ and $l_1-<!--\; ???\; -->l_2\mathrm{a}\mathrm{r}\mathrm{e}$ multiples of q. In the simple case of staircase walks, a geometrical interpretation of $Z_{m,0,l,0}({\mathrm{e}}^{\frac{2\mathrm{i}\mathrm{\pi <!--\; ??\; -->}}{q}})$ in terms of the cyclic sieving phenomenon is illustrated. Then, an expression for $Z_{m_1,m_2,l_1,l_2}(-<!--\; ???\; -->1),$ which is relevant to the Stembridge case, is proposed. Finally, the related problem of evaluating the nth moments of the Hofstadter Hamiltonian in the commensurate case is addressed. (paper)