  The aim of this paper is to extend some arithmetic results on elliptic
modular forms to the case of Hilbert modular forms. Among these results let's
mention : (1) the control of the image of the Galois representation modulo $p$,
(2) Hida's congruence criterion outside an explicit set of primes $p$, and (3)
the freeness of the integral cohomology of the Hilbert modular variety over
certain local components of the Hecke algebra and the Gorenstein property of
these local algebras.
  We study the arithmetic of the Hilbert modular forms by studying their modulo
$p$ Galois representations and our main tool is the action of the inertia
groups at the primes above $p$. In order to determine this action, we compute
the Hodge-Tate (resp. the Fontaine-Laffaille) weights of the $p$-adic (resp.
the modulo $p$) etale cohomology of the
  Hilbert modular variety. The cohomological part of our paper is inspired by
the work of Mokrane, Polo and Tilouine on the cohomology of the Siegel modular
varieties and builds upon the geometric constructions of math.NT/0212071 and
math.NT/0212072.
