Computational algorithm for reduction type of CM abelian varieties

Artyom Smirnov, Alexey Zaytsev

Research output: Contribution to journalArticlepeer-review

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 A as an abelian variety over a finite field of characteristic p, which is obtained by the reduction of A at the prime ideal. In this study, we derive an algorithm that allows us to decompose the group scheme A[p] into indecomposable quasi-polarized BT1-group schemes up to isomorphism. This can be achieved for the unramified p based on its decomposition into prime ideals in the endomorphism algebra of A. We also compute all types of these correspondences for abelian varieties with dimensions up to 5. As a consequence, we establish the relationship between the decompositions of prime p and the corresponding pairs of p-rank and a-number for an abelian variety A.

Original languageEnglish
Pages (from-to)412-429
Number of pages18
JournalJournal of Algebra
Volume542
DOIs
Publication statusPublished - 15 Jan 2020

Keywords

  • Abelian varieties
  • Group schemes

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