Invariants de classes: exemples de non-annulation en dimension supérieure

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The so-called class-invariant homomorphism ψ 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 (Néron 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 ψ 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 μₚ) 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 ψ does not vanishes on torsion points.