 
Summary: April 22, 2005 11:35 Proceedings Trim Size: 9in x 6in isog
DETECTING COMPLEX MULTIPLICATION
JEFFREY D. ACHTER
Department of Mathematics
Colorado State University
Fort Collins, CO 805231874
j.achter@colostate.edu
We give an efficient, deterministic algorithm to decide if two abelian varieties over
a number field are isogenous. From this, we derive an algorithm to compute the
endomorphism ring of an elliptic curve over a number field.
In this paper, we answer two fundamental decision problems about el
liptic curves over number fields. Specifically, we explain how to detect
whether two elliptic curves over a number field are isogenous, and how to
decide whether an elliptic curve has complex multiplication. These algo
rithms rely on Lemma 1.2, which actually applies to abelian varieties of
any dimension, and Proposition 2.1, respectively.
In each case, we answer a question about a variety over a number field by
examining its reduction at finitely many primes. At this level of generality,
such a strategy is common in algorithmic number theory. For example, a
common method for computing modular polynomials that is, bivariate
