Power Residues of Fourier Coefficients of Elliptic Curves With Complex Multiplication

Abstract

Let E be an elliptic curve over Q. For any m ≥ 1 and set of primes C (contained in the set of primes congruent to one modulo m) we define δ 1 m (E; C) as the relative density (in the set of p ∈ C which are ordinary for E) of primes p ∈ C for which the p th Fourier coefficient of E is an m th -power modulo p. In [4] it was conjectured that δ 1 m (E; C) = 1 m whenever E does not have complex multiplication and C is a set of primes defined by Galois theoretic conditions. In the present paper we extend these conjectures to the case of elliptic curves with complex multiplication; we also prove our conjectures for certain small values of m.To be more precise, fix an imaginary quadratic field K of class number one and let E denote an elliptic curve with complex multiplication by the ring of integers O K ; we write w for the order of O × K . For any divisor n of m we consider the density δ n m (E; C) of p ∈ C for which the m th power residue symbol of the p th Fourier coefficient of E modulo p is a primitive n th root of unity. We compute the density δ n m (E; C) (in terms of certain simpler densities) for any m dividing w; most of these computations were essentially done in [4], with the exception of K = Q(i) and m = 4 (which is significantly more involved). These densities are often different from the naive expectation ϕ(n) m . For general m, we conjecture that the density δ n m (E; C) differs from ϕ(n) m only to the extent that such a difference is forced upon it by its relation to the known density δ n ′ m ′ (E; C) with m ′ = (m, w) and n ′ an appropriate divisor of m ′ . We make our conjecture entirely explicit in the case that C consists of all primes congruent to one modulo m.We now outline the contents of the paper. In Section 1 we set our notation for densities and give the basic density computation coming from the Chebotarev density theorem. We give careful statements of the known results for m dividing w in Section 2; this is done most efficiently by regarding all elliptic curves with complex multiplication by O K as twists of a fixed such curve. We also give some preliminary density computations for later use in explicating our general conjectures. Those conjectures are stated in Section 3, where we also verify certain natural compatibilities. In Section 4 we give the proof, based on biquadratic reciprocity, of our conjecture in the case K = Q(i) and m = 4We emphasize that despite the essentially elementary nature of our approach for m dividing w, it remains our opinion that entirely new methods will be required to approach the general case.It is a pleasure to thank Farshid Hajir for numerous helpful conversations.