Arithmetic compactifications of the Hilbert modular varieties and Hilbert modular forms for ${\mathrm{\Gamma }}_1(c,n)$

We construct local charts for the ramified cusps of the Hilbert modular variety of level ${\mathrm{\Gamma }}_1(\mathfrak{c},\mathfrak{n})$. This allows us, following closely M. Rapoport and C.-L. Chai, to construct arithmetic toroidal and minimal compactifications of these modular varieties. Some applications to modular forms are given. See our joint work with J. Tilouine for other applications.