  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.
