Power residues of Fourier coefficients of elliptic curves with complex multiplication

2006/04/03 by Weston, Tom, Zaurova, Elena
#11F30 #11G15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0604034

Abstract

Fix m >= 1 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).

Related

• Elliptic curves and spin
• Constructing elliptic curves with a known number of points over a prime field
• Elliptic curves and the residue-counts of x²+bx+c/x modulo p
• Power residues of Fourier coefficients of modular forms
• The average exponent of elliptic curves modulo p