Alessandra Bertapelle

The Greenberg functor revisited

We extend Greenberg’s original construction to arbitrary schemes over (certain types of) local artinian rings. We then establish a number of properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor.

Sharp de Rham realization

We introduce the sharp (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the sharp de Rham realization T♯ by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. Thus we show how to provide one-dimensional sharp de Rham cohomology H♯-dR1 of algebraic varieties.

Galois sets of connected components and Weil restriction

Let k be a field, A a finite k-algebra and X a smooth A-scheme. We describe the Galois set of connected components of the Weil restriction

On deformations of

\mathfrak {R}_{A/ k}(X)

-motives

RA/k(X) in terms of the sets of connected components of the geometric fibers of X.

Weil restriction and Grothendieck''s duality conjecture

Abstract After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field, we show that it yields a full functor by making use of an algebraicity theorem of Bost.

Deligne's duality for de Rham realizations of 1‐motives

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to

On torsors under elliptic curves and Serre’s pro-algebraic structures

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