Power Residues of Fourier Coefficients of Elliptic Curves with Complex Multiplication
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
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).
T Weston and E Zaurova. "Power Residues of Fourier Coefficients of Elliptic Curves with Complex Multiplication" International Journal of Number Theory 5.1 (2009): 109-124.
Available at: http://works.bepress.com/tom_weston/6