Cohomology of Siegel varieties with p - adic integral coefficients and Applications

1 Introduction 1.1. Let G be a connected reductive group over Q. Diamond [16] and Fujiwara [27] (independently) have axiomatized the Taylor-Wiles method which allows to study some local components T m of a Hecke algebra T for G of suitable (minimal) level; when it applies, this method shows at the same time that T m is complete intersection and that some cohomology module, viewed as T-module, is locally free at m. It has been successfully applied to GL(2) /Q [68], to some quaternionic Hilbert modular cases [27], and to some inner forms of unitary groups [37]. If one tries to treat other cases, one can let the Hecke algebra act faithfully on the middle degree Betti cohomology of an associated Shimura variety; then, one of the problems to overcome is the possible presence of torsion in the cohomology modules with p-adic integral coefficients. For G = GSp(2g) (g ≥ 1), we want to explain in this paper why this torsion is not supported by maximal ideals of T which are " non-Eisenstein " and ordinary (see below for precise definitions), provided the residual characteristic p is prime to the level and greater than a natural bound. A drawback of our method is that it necessitates to assume that the existence and some local properties of the Galois representations associated to cohomological cuspidal representations on G are established. For the moment, they are proven for g ≤ 2 (see below). In his recent preprint [41], Hida explains for the same symplectic groups G how by considering only coherent cohomology, one can let the Hecke algebra act faithfully too on cohomology modules whose torsion-freeness is built-in (without assuming any conjecture). However for some applications (like the relation, for some groups G, between special values of adjoint L-functions, congruence numbers, and cardinality of adjoint Selmer groups), the use of the Betti cohomology seems indispensable. 1 1.2. Let G = GSp(2g) be the group of symplectic similitudes given by the matrix J = 0 s −s 0 , whose entries are g×g-matrices, and s is antidiagonal, with non-zero coefficients equal to 1; the standard Borel B, resp. torus T , in G consists in upper triangular matrices, resp. diagonal matrices in G. For any dominant weight λ for (G, B, T), we writê λ for its dual (that is, the dominant weight associated to the Weyl representation dual of that of λ). Let …