Every finite abelian group arises as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$, and $\mathbb{F}_5$

Abstract: We will show that every finite abelian group arises as the group of rational points of an ordinary abelian variety over a finite field with 2, 3 or 5 elements. Similar results hold over finite fields of larger cardinality. On our way to proving these results, we will view the group of rational points of an abelian variety as a module over its endomorphism ring. By describing this module structure in important cases, we obtain (a fortiori) an understanding of the underlying groups. Combining this description of structure with recent results on the cardinalities of groups of rational points of abelian varieties over finite fields, we will deduce the main theorem. This work is joint with Stefano Marseglia.

number theory

Audience: researchers in the topic

London number theory seminar

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