2006/03/31 by Jean Gillibert
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #math.AG #math.NT #msc:11Gxx #msc:14Kxx
paper · pdf · doi:10.1007/s00208-007-0084-4
published as Math. Annalen 338 (2007), 475-495 · 20 pages, LaTeX. Small changes
openalex publication_date 2007/02/01 · arxiv created 2007/10/05 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
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.