Invariants de classes : exemples de non-annulation en dimension sup\'erieure

The so-called class-invariant homomorphism $\psi$ measures the Galois module structure of torsors--under a finite flat group scheme $G$--which lie in the image of a coboundary map associated to an isogeny between (N\'eron models of) abelian varieties with kernel $G$. When the varieties are elliptic curves with semi-stable reduction and the order of $G$ is coprime to 6, is is known that the homomorphism $\psi$ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number $p>2$, of an abelian variety $A$ of dimension $p$ endowed with an isogeny (with kernel $\mu_p$) whose coboundary map is surjective. In the case when $A$ has rank zero and the $p$-part of the Picard group of the base is non-trivial, we obtain examples where $\psi$ does not vanishes on torsion points.