Computing endomorphism rings of Jacobians of genus 2 curves over finite fields

We present algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J[ℓd] for prime powers ℓd. We use these algorithms to create the first implementation of Eisentr\”ager and Lauter’s algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute p3 curves for many small primes p.