We recall first the analytic theory of the Hilbert modular varieties of level $\Gamma_1(\mathfrak{ c},\mathfrak{ n})$ and their compactifications. We construct arithmetic toroidal compactifications of the universal Hilbert-Blumenthal abelian variety (and of its fiber products), over the compactifications of the Hilbert modular variety constructed by the first author. We give several applications related to Hodge theory, Hilbert-Jacobi modular forms and Hilbert modular forms of half-integer weight.
Categories: math.NT math.AG