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

Abstract

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.

116 pages, updated version of preprint of University Paris-Nord 2000-03

Cited by in corpus (10)

• On Ihara's lemma for Hilbert Modular Varieties
• Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over ${\mathbb{Z}}_p$ for representations with p-small weights
• L-modules and the Conjecture of Rapoport and Goresky-MacPherson
• p-adic Hodge-theoretic properties of étale cohomology with mod p coefficients, and the cohomology of Shimura varieties
• On the intersection motive of certain Shimura varieties: the case of Siegel threefolds
• Lifts of Hilbert modular forms and application to modularity of abelian varieties
• Iwasawa theory for quadratic Hilbert modular forms
• Tempered currents and Deligne cohomology of Shimura varieties, with an application to ${\mathrm{GSp}}_6$
• Galois representations modulo $p$ and cohomology of Hilbert modular varieties
• Euler characteristic and cohomology of ${\mathrm{Sp}}_4(\mathbb{Z})$ with nontrivial coefficients