Power Residues of Fourier Coefficients of Elliptic Curves with Complex Multiplication

T Weston, University of Massachusetts - Amherst
E Zaurova

Article comments

This is the pre-published version harvested from ArXiv. The published version is located at http://vm-jn.wspc.com.sg/ijnt/05/0501/S1793042109001955.html

Abstract

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).