Reduction Of Abelian Varieties Research Articles

Galois representations, Mumford-Tate groups and good reduction of abelian varieties

Galois representations, Mumford-Tate groups and good reduction of abelian varieties

Étale cohomology and reduction of abelian varieties

Nous etudions les groupes de cohomologies etales associes aux varietes abeliennes. Nous obtenons des conditions necessaires et suffisantes pour qu'une variete abelienne ait une reduction semistable (ou une reduction purement additive qui devient semistable sur une extension quadratique), en termes de l'action du groupe d'inertie absolu sur les groupes de cohomologies etales a coefficients finis.

Good and stable reduction of abelian varieties

An abelian variety with sufficiently many complex multiplications has potentially good reduction; in case the residue class field is finite this was proved by Serre and Tate; in this paper we give a proof in the general case. An abelian variety has potentially stable reduction (at any discrete valuation of its field of definition); we show this theorem follows directly from the Igusa-Grothendieck orthogonality theorem.