Computing the endomorphism ring of an ordinary abelian surface over a finite field

Abstract

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker [2] which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.

The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field

“…if End Fq (A) contains the ring of integers of the maximal totally real subfield of Q(π); see [3,Lemma 4.4]. Many recent results in the algorithmic study of abelian varieties over finite fields have productively focused on the case of maximal real multiplication, including results on point counting [1,6], isogeny graphs [3,8,13], and endomorphism ring computation [16]. At the other extreme, Centeleghe and Stix have shown that the minimal order Z[π, π] is also always Gorenstein, where π is a Weil integer [4,Theorem 11].…”