Computing algorithm for reduction type of CM abelian varieties

TL;DR

This paper presents an algorithm to decompose the $p$-torsion group scheme of CM abelian varieties over finite fields, linking prime decomposition in endomorphism algebras to invariants like $p$-rank and $a$-number.

Contribution

It introduces a novel algorithm for decomposing the $p$-torsion group scheme of CM abelian varieties, connecting prime ideal decomposition to their $p$-adic invariants.

Findings

Algorithm for decomposing ${ m A}[p]$ into indecomposable ${ m BT}_1$-group schemes.

Computed types of decompositions for abelian varieties up to dimension 5.

Established relation between prime decomposition and $p$-rank, $a$-number.

Abstract

Let $\mathcal{A}$ 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 $\mathcal{A}$ 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 ${\rm BT}_1$-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 ${\rm A}$.