On the computation of the endomorphism rings of abelian surfaces

Abstract We extend the result of Bisson [4] to compute the endomorphism rings of principally polarized, absolutely simple, and ordinary abelian surfaces defined over finite fields in subexponential time in the size of the base field. The abelian surfaces covered here were excluded in [4] . This is accomplished by using techniques introduced in [17] to efficiently determine overorders for Bass and Gorenstein orders. In addition we show that the endomorphism rings of certain principally polarized, absolutely simple, and ordinary abelian surfaces are not computable in subexponential time with current methods [4] , [21] , including ours.