  Under the assumption that Galois representations associated to Siegel modular
forms exist (it is known only for genus at most 2), we show that the cohomology
with p-adic integral coefficients of Siegel Varieties, when localized at a
non-Eisenstein maximal ideal of the Hecke algebra, is torsion-free, provided
the prime p is large with the respect to the weight of the coefficient system.
The proof uses p-adic Hodge theory, the dual BGG complex modulo p in order to
compute the Hodge-Tate weights for the mod p cohomology. We apply this result
to the construction of Hida p-adic families for symplectic groups and to the
first step in the construction of a Taylor-Wiles system for these groups.
