Abelian varieties over finite fields and their groups of rational points

Abstract

Over a finite field ${\mathbb{𝔽}}_q$, abelian varieties with commutative endomorphism rings can be described by using modules over orders in étale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when $\mathrm{End}<!--FUNCTION\; APPLICATION-->(A)$ is locally Gorenstein, we show that the group structure of $A({\mathbb{𝔽}}_q)$ is determined by $\mathrm{End}<!--FUNCTION\; APPLICATION-->(A)$. In fact, the same conclusion is attained if $\mathrm{End}<!--FUNCTION\; APPLICATION-->(A)$ has local Cohen–Macaulay type at most $2$, under the additional assumption that $A$ is ordinary or $q$ is prime, although the conclusion is not true in general. Second, the description in the Gorenstein case is used to characterize cyclic isogeny classes in terms of conductor ideals. Third, going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with $N$ rational points in which every abelian group of order $N$ is realized as a group of rational points. Finally, we study when an abelian variety $A$ over ${\mathbb{𝔽}}_q$ and its dual $A^{\vee }$ satisfy or fail to satisfy several interrelated properties, namely $A\cong <!--FUNCTION\; APPLICATION-->A^{\vee }$, $A({\mathbb{𝔽}}_q)\cong <!--FUNCTION\; APPLICATION-->A^{\vee }({\mathbb{𝔽}}_q)$, and $\mathrm{End}<!--FUNCTION\; APPLICATION-->(A)=\mathrm{End}<!--FUNCTION\; APPLICATION-->(A^{\vee })$. In the process, we exhibit a sufficient condition for $A\ncong A^{\vee }$ involving the local Cohen–Macaulay type of $\mathrm{End}<!--FUNCTION\; APPLICATION-->(A)$. In particular, such an abelian variety $A$ is not a Jacobian, or even principally polarizable.