Springer, C; (2021) The structure of the group of rational points of an abelian variety over a finite field. European Journal of Mathematics , 7 (3) pp. 1124-1136. 10.1007/s40879-021-00460-1.

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Abstract

Let A be a simple abelian variety of dimension g defined over a finite field Fq with Frobenius endomorphism π. This paper describes the structure of the group of rational points A(Fqn), for all n>1, as a module over the ring R of endomorphisms which are defined over Fq, under certain technical conditions. If [InlineEquation not available: see fulltext.] and R is a Gorenstein ring, then [InlineEquation not available: see fulltext.]. This includes the case when A is ordinary and has maximal real multiplication. Otherwise, if Z is the center of R and [InlineEquation not available: see fulltext.] is the product of invertible prime ideals in Z, then [InlineEquation not available: see fulltext.] where [InlineEquation not available: see fulltext.]. Finally, we deduce the structure of A(F¯ q) as a module over R under similar conditions. These results generalize results of Lenstra for elliptic curves.

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