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

*David Freeman and Kristin Lauter*

**Abstract: **We present probabilistic 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 algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J[l^d] for prime powers l^d. We use these algorithms to create the first implementation of Eisentrager and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem, 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.

**Category / Keywords: **implementation / CM method, hyperelliptic curves, jacobians, genus 2, point counting

**Publication Info: **Proceedings of SAGA 2007, Tahiti (to appear)

**Date: **received 10 Jan 2007, last revised 30 May 2007

**Contact author: **dfreeman at math berkeley edu

**Available format(s): **PDF | BibTeX Citation

**Note: **Revised version, incorporating reader feedback.

**Version: **20070530:231254 (All versions of this report)

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