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April 22, 2005 11:35 Proceedings Trim Size: 9in x 6in isog DETECTING COMPLEX MULTIPLICATION
 

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 80523-1874
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

  

Source: Achter, Jeff - Department of Mathematics, Colorado State University

 

Collections: Environmental Sciences and Ecology; Mathematics