arXiv subjects

J. Tilouine

Publications and source records attributed to J. Tilouine.

3 recordsLinked to original sources

arXiv2023

On the cohomology of GL(N) and adjoint Selmer groups

We prove -under certain conditions (local-global compatibility and vanishing of integral cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of GL(N) over a CM field and we relate the localization of penultimate non vanishing cuspidal cohomology group for a locally symmetric space to the Selmer group of the Tate dual of the adjoint representation. More precisely we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as its first Tate-Shafarevich group. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. We generalize this to Hida families as well.

math.NT↗

arXiv2000

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

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.

math.AG↗

arXiv2000

Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\Z_p$ for representations with p-small weights

We show that for $p$small highest weight $λ$, 1) there is a $\Z_p$-integral version of the Bernstein-Gelfand-Gelfand complex, still a direct summand subcomplex of the standard complex for $V(λ)$ 2) Similarly, a $\Z_p$-integral (as well as a mod. p) version of Kostant formula holds true. This paper is a companion paper to the one by Mokrane-Tilouine (AG, subm. 12/12/00), where these results are requested.

math.RT↗