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

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

2007-01-10

arxiv.org/abs/math/0701305v2

Mathematics

Number Theory

Revised version, with minor corrections and incorporating reader comments. Proposition 3.7 and Lemma 6.5 are new. To appear in

Scientific paper

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 $J[\ell^d]$ for prime powers $\ell^d$. 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 $p^3$ curves for many small primes $p$.

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