Power residues of Fourier coefficients of elliptic curves with complex multiplication

Details Power residues of Fourier coefficients of elliptic curves with complex multiplication Power residues of Fourier coefficients of elliptic curves with complex multiplication

2006-04-03

arxiv.org/abs/math/0604034v1

Mathematics

Number Theory

Scientific paper

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).

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