2004/12/29 by Mikhail V. Bondarko, Bondarko, M. V.
Computer Science · Mathematics · #11G10 #11S31 #14G20 #14L05 #14L15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.math/0412521
openalex publication_date 2004/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify group schemes in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes; in particular, the minimal dimension of a formal group law that contains S as a closed subgroup is equal to the minimal number of generators for the affine algebra of S. As an application the following reduction criteria for Abelian varieties are proved. Let K be a mixed characteristic local field, let its residue field have characteristic p, L be a finite extension of K, let \mathfrakOK⊂\mathfrakOL be their rings of integers. Let e be the absolute ramification index of L, s=[logp(pe/(p-1))], e0 be the ramification index of L/K, l=2s+vp(e0)+1. For a finite flat commutative \mathfrakOL-group scheme H we denote the \mathfrakOL-dual of the module J/J2 by TH. Here J is the augmentation ideal of the affine algebra of H. Let V be an m-dimensional Abelian variety over K. Suppose that V has semistable reduction over L. \begintheor V has semistable reduction over K if and only if for some group scheme H over \mathfrakOK there exist embeddings of HK into Ker[pl]V,K, and of (\mathfrakOL/pl\mathfrakOL)m into TH_\ol. \endtheor This criterion has a very nice-looking version in the ordinary reduction case. \begintheor V has ordinary reduction over K if and only if for some HK⊂ Ker[pl]V,K and M unramified over K we have HM≅ (μpl,M)m. Here μ denotes the group scheme of roots of unity.\endtheor