Computing algorithm for reduction type of CM abelian varieties

2018/09/27 by Smirnov, Artyom, Zaytsev, Alexey
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1809.10368

Abstract

Let A be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number p. Denote by \rm A an abelian variety over a finite field of characteristic p, obtained by the reduction of A at the prime ideal. In this paper we derive an algorithm which allows to decompose the group scheme \rm A[p] into indecomposable quasi-polarized \rm BT₁-group schemes. This can be done for the unramified p on the basis of its decomposition into prime ideals in the endomorphism algebra of \rm A. We also compute all types of such correspondence for abelian varieties of dimension up to 5. As a consequence we establish the relation between the decompositions of prime p and the corresponding pairs of p-rank and a-number of an abelian variety \rm A.

Related

• Decomposing abelian varieties into simple factors: algorithms and applications
• Generalization of Deuring Reduction Theorem
• Weyl Groups and Abelian Varieties
• The support problem for abelian varieties
• Abelian Varieties with p-rank Zero