Lattices in Tate modules

Refining a theorem of Zarhin, we prove that, given a g-dimensional abelian variety X and an endomorphism u of X, there exists a matrix A∈M2g(ℤ) such that each Tate module TℓX has a ℤℓ-basis on which the action of u is given by A, and similarly for the covariant Dieudonné module if over a perfect field of characteristic p.

Dieudonné theory via cohomology of classifying stacks

We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$ ) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.

Kernels in the Category of Formal Group Laws

Fontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic p with the aid of triples consisting of the module of logarithms, the Dieudonné module, and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of ℚp and of a totally ramiûed extension of degree less or equal than p. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from L-series.